# Loading Libraries
library(tidyverse)
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library(DataExplorer)
library(openxlsx)
library(nortest)
library(car)
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library(MASS)
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library(lmtest)
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library(olsrr)
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library(Ecdat)
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library(Amelia)
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## ## 
## ## Amelia II: Multiple Imputation
## ## (Version 1.8.0, built: 2021-05-26)
## ## Copyright (C) 2005-2022 James Honaker, Gary King and Matthew Blackwell
## ## Refer to http://gking.harvard.edu/amelia/ for more information
## ##
library(missRanger)
library(ggplot2)
library(plotly)
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Importing the dataset

df <- read.csv(choose.files(), header = T)
head(df)
##       Country Year     Status Life.expectancy Adult.Mortality infant.deaths
## 1 Afghanistan 2015 Developing            65.0             263            62
## 2 Afghanistan 2014 Developing            59.9             271            64
## 3 Afghanistan 2013 Developing            59.9             268            66
## 4 Afghanistan 2012 Developing            59.5             272            69
## 5 Afghanistan 2011 Developing            59.2             275            71
## 6 Afghanistan 2010 Developing            58.8             279            74
##   Alcohol percentage.expenditure Hepatitis.B Measles  BMI under.five.deaths
## 1    0.01              71.279624          65    1154 19.1                83
## 2    0.01              73.523582          62     492 18.6                86
## 3    0.01              73.219243          64     430 18.1                89
## 4    0.01              78.184215          67    2787 17.6                93
## 5    0.01               7.097109          68    3013 17.2                97
## 6    0.01              79.679367          66    1989 16.7               102
##   Polio Total.expenditure Diphtheria HIV.AIDS       GDP Population
## 1     6              8.16         65      0.1 584.25921   33736494
## 2    58              8.18         62      0.1 612.69651     327582
## 3    62              8.13         64      0.1 631.74498   31731688
## 4    67              8.52         67      0.1 669.95900    3696958
## 5    68              7.87         68      0.1  63.53723    2978599
## 6    66              9.20         66      0.1 553.32894    2883167
##   thinness..1.19.years thinness.5.9.years Income.composition.of.resources
## 1                 17.2               17.3                           0.479
## 2                 17.5               17.5                           0.476
## 3                 17.7               17.7                           0.470
## 4                 17.9               18.0                           0.463
## 5                 18.2               18.2                           0.454
## 6                 18.4               18.4                           0.448
##   Schooling
## 1      10.1
## 2      10.0
## 3       9.9
## 4       9.8
## 5       9.5
## 6       9.2

Data Preprocessing

Checking the structure of the dataset

str(df)
## 'data.frame':    2938 obs. of  22 variables:
##  $ Country                        : chr  "Afghanistan" "Afghanistan" "Afghanistan" "Afghanistan" ...
##  $ Year                           : int  2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 ...
##  $ Status                         : chr  "Developing" "Developing" "Developing" "Developing" ...
##  $ Life.expectancy                : num  65 59.9 59.9 59.5 59.2 58.8 58.6 58.1 57.5 57.3 ...
##  $ Adult.Mortality                : int  263 271 268 272 275 279 281 287 295 295 ...
##  $ infant.deaths                  : int  62 64 66 69 71 74 77 80 82 84 ...
##  $ Alcohol                        : num  0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.03 0.02 0.03 ...
##  $ percentage.expenditure         : num  71.3 73.5 73.2 78.2 7.1 ...
##  $ Hepatitis.B                    : int  65 62 64 67 68 66 63 64 63 64 ...
##  $ Measles                        : int  1154 492 430 2787 3013 1989 2861 1599 1141 1990 ...
##  $ BMI                            : num  19.1 18.6 18.1 17.6 17.2 16.7 16.2 15.7 15.2 14.7 ...
##  $ under.five.deaths              : int  83 86 89 93 97 102 106 110 113 116 ...
##  $ Polio                          : int  6 58 62 67 68 66 63 64 63 58 ...
##  $ Total.expenditure              : num  8.16 8.18 8.13 8.52 7.87 9.2 9.42 8.33 6.73 7.43 ...
##  $ Diphtheria                     : int  65 62 64 67 68 66 63 64 63 58 ...
##  $ HIV.AIDS                       : num  0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 ...
##  $ GDP                            : num  584.3 612.7 631.7 670 63.5 ...
##  $ Population                     : num  33736494 327582 31731688 3696958 2978599 ...
##  $ thinness..1.19.years           : num  17.2 17.5 17.7 17.9 18.2 18.4 18.6 18.8 19 19.2 ...
##  $ thinness.5.9.years             : num  17.3 17.5 17.7 18 18.2 18.4 18.7 18.9 19.1 19.3 ...
##  $ Income.composition.of.resources: num  0.479 0.476 0.47 0.463 0.454 0.448 0.434 0.433 0.415 0.405 ...
##  $ Schooling                      : num  10.1 10 9.9 9.8 9.5 9.2 8.9 8.7 8.4 8.1 ...

Checking for the missing values

sum(is.na(df))
## [1] 2563
names(which(colSums(is.na(df))>0))
##  [1] "Life.expectancy"                 "Adult.Mortality"                
##  [3] "Alcohol"                         "Hepatitis.B"                    
##  [5] "BMI"                             "Polio"                          
##  [7] "Total.expenditure"               "Diphtheria"                     
##  [9] "GDP"                             "Population"                     
## [11] "thinness..1.19.years"            "thinness.5.9.years"             
## [13] "Income.composition.of.resources" "Schooling"

There are 2563 missing values in 14 different columns.

Imputing the missing values

df_imp <- missRanger(df, pmm.k = 3, seed = 153)
## 
## Missing value imputation by random forests
## 
##   Variables to impute:       Life.expectancy, Adult.Mortality, Alcohol, Hepatitis.B, BMI, Polio, Total.expenditure, Diphtheria, GDP, Population, thinness..1.19.years, thinness.5.9.years, Income.composition.of.resources, Schooling
##   Variables used to impute:  Country, Year, Status, Life.expectancy, Adult.Mortality, infant.deaths, Alcohol, percentage.expenditure, Hepatitis.B, Measles, BMI, under.five.deaths, Polio, Total.expenditure, Diphtheria, HIV.AIDS, GDP, Population, thinness..1.19.years, thinness.5.9.years, Income.composition.of.resources, Schooling
## iter 1:  ..............
## iter 2:  ..............
## iter 3:  ..............
## iter 4:  ..............
head(df_imp)
##       Country Year     Status Life.expectancy Adult.Mortality infant.deaths
## 1 Afghanistan 2015 Developing            65.0             263            62
## 2 Afghanistan 2014 Developing            59.9             271            64
## 3 Afghanistan 2013 Developing            59.9             268            66
## 4 Afghanistan 2012 Developing            59.5             272            69
## 5 Afghanistan 2011 Developing            59.2             275            71
## 6 Afghanistan 2010 Developing            58.8             279            74
##   Alcohol percentage.expenditure Hepatitis.B Measles  BMI under.five.deaths
## 1    0.01              71.279624          65    1154 19.1                83
## 2    0.01              73.523582          62     492 18.6                86
## 3    0.01              73.219243          64     430 18.1                89
## 4    0.01              78.184215          67    2787 17.6                93
## 5    0.01               7.097109          68    3013 17.2                97
## 6    0.01              79.679367          66    1989 16.7               102
##   Polio Total.expenditure Diphtheria HIV.AIDS       GDP Population
## 1     6              8.16         65      0.1 584.25921   33736494
## 2    58              8.18         62      0.1 612.69651     327582
## 3    62              8.13         64      0.1 631.74498   31731688
## 4    67              8.52         67      0.1 669.95900    3696958
## 5    68              7.87         68      0.1  63.53723    2978599
## 6    66              9.20         66      0.1 553.32894    2883167
##   thinness..1.19.years thinness.5.9.years Income.composition.of.resources
## 1                 17.2               17.3                           0.479
## 2                 17.5               17.5                           0.476
## 3                 17.7               17.7                           0.470
## 4                 17.9               18.0                           0.463
## 5                 18.2               18.2                           0.454
## 6                 18.4               18.4                           0.448
##   Schooling
## 1      10.1
## 2      10.0
## 3       9.9
## 4       9.8
## 5       9.5
## 6       9.2
range(df_imp$Life.expectancy)
## [1] 36.3 89.0
sum(is.na(df_imp))
## [1] 0

Data Cleaning

Creating a new variable with all the columns that are required for the analysis and model building.

Removing “Country” and “Year”, because it has too many variables and time series which doesn’t really help in predicting life expectancy.

Using another variable for Categorical Analysis

life_selected <- df_imp %>% 
                          mutate(Hepatitis.B = ifelse(Hepatitis.B < 90, "<90% Covered", ">=90% Covered"),
                                 Polio = ifelse(Polio < 90, "<90% Covered", ">=90% Covered"),
                                 Diphtheria = ifelse(Diphtheria < 90, "<90% Covered", ">=90% Covered"),
                                 Hepatitis.B = as.factor(Hepatitis.B),
                                 Polio = as.factor(Polio),
                                 Diphtheria = as.factor(Diphtheria))
life_selected$Country <- NULL
life_selected$Year <- NULL

str(life_selected)
## 'data.frame':    2938 obs. of  20 variables:
##  $ Status                         : chr  "Developing" "Developing" "Developing" "Developing" ...
##  $ Life.expectancy                : num  65 59.9 59.9 59.5 59.2 58.8 58.6 58.1 57.5 57.3 ...
##  $ Adult.Mortality                : int  263 271 268 272 275 279 281 287 295 295 ...
##  $ infant.deaths                  : int  62 64 66 69 71 74 77 80 82 84 ...
##  $ Alcohol                        : num  0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.03 0.02 0.03 ...
##  $ percentage.expenditure         : num  71.3 73.5 73.2 78.2 7.1 ...
##  $ Hepatitis.B                    : Factor w/ 2 levels "<90% Covered",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ Measles                        : int  1154 492 430 2787 3013 1989 2861 1599 1141 1990 ...
##  $ BMI                            : num  19.1 18.6 18.1 17.6 17.2 16.7 16.2 15.7 15.2 14.7 ...
##  $ under.five.deaths              : int  83 86 89 93 97 102 106 110 113 116 ...
##  $ Polio                          : Factor w/ 2 levels "<90% Covered",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ Total.expenditure              : num  8.16 8.18 8.13 8.52 7.87 9.2 9.42 8.33 6.73 7.43 ...
##  $ Diphtheria                     : Factor w/ 2 levels "<90% Covered",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ HIV.AIDS                       : num  0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 ...
##  $ GDP                            : num  584.3 612.7 631.7 670 63.5 ...
##  $ Population                     : num  33736494 327582 31731688 3696958 2978599 ...
##  $ thinness..1.19.years           : num  17.2 17.5 17.7 17.9 18.2 18.4 18.6 18.8 19 19.2 ...
##  $ thinness.5.9.years             : num  17.3 17.5 17.7 18 18.2 18.4 18.7 18.9 19.1 19.3 ...
##  $ Income.composition.of.resources: num  0.479 0.476 0.47 0.463 0.454 0.448 0.434 0.433 0.415 0.405 ...
##  $ Schooling                      : num  10.1 10 9.9 9.8 9.5 9.2 8.9 8.7 8.4 8.1 ...
life_selected$Status <- as.factor(life_selected$Status)

str(life_selected)
## 'data.frame':    2938 obs. of  20 variables:
##  $ Status                         : Factor w/ 2 levels "Developed","Developing": 2 2 2 2 2 2 2 2 2 2 ...
##  $ Life.expectancy                : num  65 59.9 59.9 59.5 59.2 58.8 58.6 58.1 57.5 57.3 ...
##  $ Adult.Mortality                : int  263 271 268 272 275 279 281 287 295 295 ...
##  $ infant.deaths                  : int  62 64 66 69 71 74 77 80 82 84 ...
##  $ Alcohol                        : num  0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.03 0.02 0.03 ...
##  $ percentage.expenditure         : num  71.3 73.5 73.2 78.2 7.1 ...
##  $ Hepatitis.B                    : Factor w/ 2 levels "<90% Covered",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ Measles                        : int  1154 492 430 2787 3013 1989 2861 1599 1141 1990 ...
##  $ BMI                            : num  19.1 18.6 18.1 17.6 17.2 16.7 16.2 15.7 15.2 14.7 ...
##  $ under.five.deaths              : int  83 86 89 93 97 102 106 110 113 116 ...
##  $ Polio                          : Factor w/ 2 levels "<90% Covered",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ Total.expenditure              : num  8.16 8.18 8.13 8.52 7.87 9.2 9.42 8.33 6.73 7.43 ...
##  $ Diphtheria                     : Factor w/ 2 levels "<90% Covered",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ HIV.AIDS                       : num  0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 ...
##  $ GDP                            : num  584.3 612.7 631.7 670 63.5 ...
##  $ Population                     : num  33736494 327582 31731688 3696958 2978599 ...
##  $ thinness..1.19.years           : num  17.2 17.5 17.7 17.9 18.2 18.4 18.6 18.8 19 19.2 ...
##  $ thinness.5.9.years             : num  17.3 17.5 17.7 18 18.2 18.4 18.7 18.9 19.1 19.3 ...
##  $ Income.composition.of.resources: num  0.479 0.476 0.47 0.463 0.454 0.448 0.434 0.433 0.415 0.405 ...
##  $ Schooling                      : num  10.1 10 9.9 9.8 9.5 9.2 8.9 8.7 8.4 8.1 ...

Exploratory Data Analysis

Categorical Variables

Checking the data distribution of Life.expectancy among all of the Categorical Variables

Status
life_selected %>% 
                 group_by(Status) %>% 
                 summarise(count = n()) %>% 
                 mutate(percentage = paste0(round(count/sum(count)*100, 2), "%"))
## # A tibble: 2 x 3
##   Status     count percentage
##   <fct>      <int> <chr>     
## 1 Developed    512 17.43%    
## 2 Developing  2426 82.57%
plot1 <-  ggplot(life_selected, aes(x=Status, y = Life.expectancy, fill = Status)) +
                geom_boxplot() +
                scale_fill_manual(values=c("green3", "darkorange")) +
                labs(x = "Development Status", y = "Life Expectancy (Age)") +
                theme(legend.position = "none")


ggplotly(plot1)
summary(aov(Life.expectancy ~ Status, data = life_selected))
##               Df Sum Sq Mean Sq F value Pr(>F)    
## Status         1  61418   61418   881.8 <2e-16 ***
## Residuals   2936 204499      70                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  • The number of developing nations is significantly higher than the number of developed nations in these observations.
  • According to the Development Status, it was evident that the Developed Countries have a greater Life Expectancy Distribution, with a Significant Median Distance. Additionally, even though there are some developing country outliers, we will keep them for the time being because they have low leverage.
  • We can infer that there are substantial disparities in life expectancy between developed and developing countries because the p-value from the ANOVA analysis is less than the significance level of 0.05.
Hepatitis B Coverage
life_selected %>% 
                 group_by(Hepatitis.B) %>% 
                 summarise(count = n()) %>% 
                 mutate(percentage = paste0(round(count/sum(count)*100, 2), "%"))
## # A tibble: 2 x 3
##   Hepatitis.B   count percentage
##   <fct>         <int> <chr>     
## 1 <90% Covered   1504 51.19%    
## 2 >=90% Covered  1434 48.81%
plot2 <-  ggplot(life_selected, aes(x=Hepatitis.B, y = Life.expectancy, fill = Hepatitis.B)) +
                geom_boxplot() +
                scale_fill_manual(values=c("green3", "darkorange")) +
                labs(x = "Hepatitis B Coverage", y = "Life Expectancy (Age)") +
                theme(legend.position = "none")


ggplotly(plot2)
summary(aov(Life.expectancy ~ Hepatitis.B, data = life_selected))
##               Df Sum Sq Mean Sq F value Pr(>F)    
## Hepatitis.B    1  41593   41593   544.4 <2e-16 ***
## Residuals   2936 224324      76                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  • Surprisingly, half of the findings are from countries where hepatitis.B vaccination coverage is less than 90%.
  • Greater life expectancy is associated with countries that cover their hepatitis B vaccination at a rate of 90% or higher, with a significant median gap. Even though there are few outliers among the developing nations, the majority of them have low leverage, so we will preserve it for the time being.
  • We can infer that there are significant differences in Life Expectancy across the groups in Hepatitis B Coverage because the p-value is less than the significance level of 0.05.
Polio Coverage
life_selected %>% 
                 group_by(Polio) %>% 
                 summarise(count = n()) %>% 
                 mutate(percentage = paste0(round(count/sum(count)*100, 2), "%"))
## # A tibble: 2 x 3
##   Polio         count percentage
##   <fct>         <int> <chr>     
## 1 <90% Covered   1248 42.48%    
## 2 >=90% Covered  1690 57.52%
plot3 <-  ggplot(life_selected, aes(x=Polio, y = Life.expectancy, fill = Polio)) +
                geom_boxplot() +
                scale_fill_manual(values=c("green3", "darkorange")) +
                labs(x = "Polio Coverage", y = "Life Expectancy (Age)") +
                theme(legend.position = "none")


ggplotly(plot3)
summary(aov(Life.expectancy ~ Polio, data = life_selected))
##               Df Sum Sq Mean Sq F value Pr(>F)    
## Polio          1  85316   85316    1387 <2e-16 ***
## Residuals   2936 180600      62                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  • Greater Polio Coverage is better than Hepatitis B Coverage.
  • Greater life expectancy is associated with countries that have polio vaccination coverage rates of 90% or higher, with a significant median distance. Polio does not have the same high outliers as Hepatitis.B. Even though there are few outliers among the developing nations, the majority of them have low leverage, so we will preserve it for the time being.
  • We can draw the conclusion that there are significant differences in life expectancy between the groups in polio coverage because the p-value of the ANOVA Analysis is less than the significance level of 0.05.
Diphteria
life_selected %>% 
                 group_by(Diphtheria) %>% 
                 summarise(count = n()) %>% 
                 mutate(percentage = paste0(round(count/sum(count)*100, 2), "%"))
## # A tibble: 2 x 3
##   Diphtheria    count percentage
##   <fct>         <int> <chr>     
## 1 <90% Covered   1262 42.95%    
## 2 >=90% Covered  1676 57.05%
plot4 <-  ggplot(life_selected, aes(x=Diphtheria, y = Life.expectancy, fill = Diphtheria)) +
                geom_boxplot() +
                scale_fill_manual(values=c("green3", "darkorange")) +
                labs(x = "Diphtheria Coverage", y = "Life Expectancy (Age)") +
                theme(legend.position = "none")


ggplotly(plot4)
summary(aov(Life.expectancy ~ Diphtheria, data = life_selected))
##               Df Sum Sq Mean Sq F value Pr(>F)    
## Diphtheria     1  85302   85302    1387 <2e-16 ***
## Residuals   2936 180615      62                   
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
  • In terms of the number of countries, diphtheria coverage is comparable to polio coverage.
  • The spread of diphtheria and polio are somewhat comparable. It can mean that Diphtheria and Polio vaccinations are administered simultaneously.
  • We can draw the conclusion that there are significant differences in life expectancy between the groups in diphtheria coverage because the p-value of the ANOVA Analysis is less than the significance level of 0.05.

Association between Categorical Variables

Development Status vs Hepatitis B Coverage
library(ggmosaic)
plot5 <-  ggplot(life_selected) +
            geom_mosaic(aes(x = product(Status), fill=Hepatitis.B)) +
            labs(x = NULL, y = NULL) +
            scale_fill_manual(values=c("green3", "darkorange")) +
            theme(legend.position = "none")

ggplotly(plot5) 
## Warning: `unite_()` was deprecated in tidyr 1.2.0.
## Please use `unite()` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was generated.
chisq.test(table(life_selected$Status, life_selected$Hepatitis.B))
## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  table(life_selected$Status, life_selected$Hepatitis.B)
## X-squared = 54.105, df = 1, p-value = 1.901e-13
  • The majority of developed nations have higher vaccination rates for hepatitis B.
  • By using the chi-square test, we can see compelling evidence that developed and developing countries typically have different coverage rates for the hepatitis.B vaccine.
Development Status vs Polio Coverage
plot6 <-  ggplot(life_selected) +
            geom_mosaic(aes(x = product(Status), fill=Polio)) +
            labs(x = NULL, y = NULL) +
            scale_fill_manual(values=c("green3", "darkorange")) +
            theme(legend.position = "none")

ggplotly(plot6) 
chisq.test(table(life_selected$Status, life_selected$Polio))
## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  table(life_selected$Status, life_selected$Polio)
## X-squared = 269.93, df = 1, p-value < 2.2e-16
  • Developed nations have much higher inoculation rates against polio
  • By using the chi-square test, we can observe compelling evidence that developed and developing countries typically have different vaccine coverage rates for polio.
Development Status vs Diphtheria Coverage
plot7 <-  ggplot(life_selected) +
            geom_mosaic(aes(x = product(Status), fill=Diphtheria)) +
            labs(x = NULL, y = NULL) +
            scale_fill_manual(values=c("green3", "darkorange")) +
            theme(legend.position = "none")

ggplotly(plot7) 
chisq.test(table(life_selected$Status, life_selected$Diphtheria))
## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  table(life_selected$Status, life_selected$Diphtheria)
## X-squared = 286.99, df = 1, p-value < 2.2e-16
  • The relationship between Diphtheria and Polio is comparable to earlier discoveries. On the following test, we’ll determine if we just require one of them.
Using the dataframe to split the categorical variables to numerical variables
life_exp <- df_imp

life_exp$Developed <- ifelse(life_exp$Status == "Developed", 1, 0)
life_exp$Developing <- ifelse(life_exp$Status == "Developing", 1, 0)
life_exp$Hepatitis.B_below90 <- ifelse(life_exp$Hepatitis.B < 90, 1, 0)
life_exp$Hepatitis.B_above90 <- ifelse(life_exp$Hepatitis.B >= 90, 1, 0)
life_exp$Polio_below90 <- ifelse(life_exp$Polio < 90, 1, 0)
life_exp$Polio_above90 <- ifelse(life_exp$Polio >= 90, 1, 0)
life_exp$Diphtheria_below90 <- ifelse(life_exp$Diphtheria < 90, 1, 0)
life_exp$Diphtheria_above90 <- ifelse(life_exp$Diphtheria >= 90, 1, 0)

Dropping the Unwanted columns

life_exp <- life_exp[c(4:6, 10:12, 14, 16:30)]

head(life_exp)
##   Life.expectancy Adult.Mortality infant.deaths Measles  BMI under.five.deaths
## 1            65.0             263            62    1154 19.1                83
## 2            59.9             271            64     492 18.6                86
## 3            59.9             268            66     430 18.1                89
## 4            59.5             272            69    2787 17.6                93
## 5            59.2             275            71    3013 17.2                97
## 6            58.8             279            74    1989 16.7               102
##   Total.expenditure HIV.AIDS       GDP Population thinness..1.19.years
## 1              8.16      0.1 584.25921   33736494                 17.2
## 2              8.18      0.1 612.69651     327582                 17.5
## 3              8.13      0.1 631.74498   31731688                 17.7
## 4              8.52      0.1 669.95900    3696958                 17.9
## 5              7.87      0.1  63.53723    2978599                 18.2
## 6              9.20      0.1 553.32894    2883167                 18.4
##   thinness.5.9.years Income.composition.of.resources Schooling Developed
## 1               17.3                           0.479      10.1         0
## 2               17.5                           0.476      10.0         0
## 3               17.7                           0.470       9.9         0
## 4               18.0                           0.463       9.8         0
## 5               18.2                           0.454       9.5         0
## 6               18.4                           0.448       9.2         0
##   Developing Hepatitis.B_below90 Hepatitis.B_above90 Polio_below90
## 1          1                   1                   0             1
## 2          1                   1                   0             1
## 3          1                   1                   0             1
## 4          1                   1                   0             1
## 5          1                   1                   0             1
## 6          1                   1                   0             1
##   Polio_above90 Diphtheria_below90 Diphtheria_above90
## 1             0                  1                  0
## 2             0                  1                  0
## 3             0                  1                  0
## 4             0                  1                  0
## 5             0                  1                  0
## 6             0                  1                  0
str(life_exp)
## 'data.frame':    2938 obs. of  22 variables:
##  $ Life.expectancy                : num  65 59.9 59.9 59.5 59.2 58.8 58.6 58.1 57.5 57.3 ...
##  $ Adult.Mortality                : int  263 271 268 272 275 279 281 287 295 295 ...
##  $ infant.deaths                  : int  62 64 66 69 71 74 77 80 82 84 ...
##  $ Measles                        : int  1154 492 430 2787 3013 1989 2861 1599 1141 1990 ...
##  $ BMI                            : num  19.1 18.6 18.1 17.6 17.2 16.7 16.2 15.7 15.2 14.7 ...
##  $ under.five.deaths              : int  83 86 89 93 97 102 106 110 113 116 ...
##  $ Total.expenditure              : num  8.16 8.18 8.13 8.52 7.87 9.2 9.42 8.33 6.73 7.43 ...
##  $ HIV.AIDS                       : num  0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 ...
##  $ GDP                            : num  584.3 612.7 631.7 670 63.5 ...
##  $ Population                     : num  33736494 327582 31731688 3696958 2978599 ...
##  $ thinness..1.19.years           : num  17.2 17.5 17.7 17.9 18.2 18.4 18.6 18.8 19 19.2 ...
##  $ thinness.5.9.years             : num  17.3 17.5 17.7 18 18.2 18.4 18.7 18.9 19.1 19.3 ...
##  $ Income.composition.of.resources: num  0.479 0.476 0.47 0.463 0.454 0.448 0.434 0.433 0.415 0.405 ...
##  $ Schooling                      : num  10.1 10 9.9 9.8 9.5 9.2 8.9 8.7 8.4 8.1 ...
##  $ Developed                      : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ Developing                     : num  1 1 1 1 1 1 1 1 1 1 ...
##  $ Hepatitis.B_below90            : num  1 1 1 1 1 1 1 1 1 1 ...
##  $ Hepatitis.B_above90            : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ Polio_below90                  : num  1 1 1 1 1 1 1 1 1 1 ...
##  $ Polio_above90                  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ Diphtheria_below90             : num  1 1 1 1 1 1 1 1 1 1 ...
##  $ Diphtheria_above90             : num  0 0 0 0 0 0 0 0 0 0 ...
Correlation
life_cor <- cor(life_exp)
life_cor
##                                 Life.expectancy Adult.Mortality infant.deaths
## Life.expectancy                      1.00000000    -0.696626466   -0.19692975
## Adult.Mortality                     -0.69662647     1.000000000    0.07931261
## infant.deaths                       -0.19692975     0.079312611    1.00000000
## Measles                             -0.15790099     0.031673459    0.50112834
## BMI                                  0.57134224    -0.393065148   -0.22758957
## under.five.deaths                   -0.22288775     0.094697101    0.99662888
## Total.expenditure                    0.23044622    -0.127919303   -0.13102225
## HIV.AIDS                            -0.55670595     0.523993429    0.02523132
## GDP                                  0.44917692    -0.295235102   -0.10642924
## Population                          -0.03294244    -0.001989446    0.55726765
## thinness..1.19.years                -0.48047136     0.307818627    0.46469284
## thinness.5.9.years                  -0.47437834     0.312750080    0.47038449
## Income.composition.of.resources      0.73649241    -0.474912519   -0.16470019
## Schooling                            0.76208016    -0.472390497   -0.21271968
## Developed                            0.48058927    -0.313682017   -0.11225158
## Developing                          -0.48058927     0.313682017    0.11225158
## Hepatitis.B_below90                 -0.39549144     0.262109092    0.17495190
## Hepatitis.B_above90                  0.39549144    -0.262109092   -0.17495190
## Polio_below90                       -0.56642658     0.366536927    0.21411536
## Polio_above90                        0.56642658    -0.366536927   -0.21411536
## Diphtheria_below90                  -0.56637789     0.363056541    0.21238743
## Diphtheria_above90                   0.56637789    -0.363056541   -0.21238743
##                                     Measles         BMI under.five.deaths
## Life.expectancy                 -0.15790099  0.57134224       -0.22288775
## Adult.Mortality                  0.03167346 -0.39306515        0.09469710
## infant.deaths                    0.50112834 -0.22758957        0.99662888
## Measles                          1.00000000 -0.17426596        0.50780871
## BMI                             -0.17426596  1.00000000       -0.23834974
## under.five.deaths                0.50780871 -0.23834974        1.00000000
## Total.expenditure               -0.10698225  0.25450586       -0.13265932
## HIV.AIDS                         0.03089872 -0.24243839        0.03806151
## GDP                             -0.07355282  0.29406686       -0.10999641
## Population                       0.26715586 -0.06966492        0.54491457
## thinness..1.19.years             0.22349730 -0.53560160        0.46715303
## thinness.5.9.years               0.21990855 -0.54218736        0.47168144
## Income.composition.of.resources -0.15351757  0.52084467       -0.18385354
## Schooling                       -0.16526449  0.56066078       -0.22954523
## Developed                       -0.07695507  0.31377031       -0.11519509
## Developing                       0.07695507 -0.31377031        0.11519509
## Hepatitis.B_below90              0.12264181 -0.23793027        0.18566031
## Hepatitis.B_above90             -0.12264181  0.23793027       -0.18566031
## Polio_below90                    0.15136310 -0.33780111        0.22620022
## Polio_above90                   -0.15136310  0.33780111       -0.22620022
## Diphtheria_below90               0.14053796 -0.33316029        0.22474158
## Diphtheria_above90              -0.14053796  0.33316029       -0.22474158
##                                 Total.expenditure     HIV.AIDS         GDP
## Life.expectancy                       0.230446220 -0.556705954  0.44917692
## Adult.Mortality                      -0.127919303  0.523993429 -0.29523510
## infant.deaths                        -0.131022249  0.025231318 -0.10642924
## Measles                              -0.106982247  0.030898718 -0.07355282
## BMI                                   0.254505864 -0.242438392  0.29406686
## under.five.deaths                    -0.132659321  0.038061512 -0.10999641
## Total.expenditure                     1.000000000 -0.005493024  0.14162395
## HIV.AIDS                             -0.005493024  1.000000000 -0.12760549
## GDP                                   0.141623948 -0.127605489  1.00000000
## Population                           -0.057854707 -0.020012844 -0.03156714
## thinness..1.19.years                 -0.290820063  0.201795799 -0.28173976
## thinness.5.9.years                   -0.297970679  0.204798392 -0.28534864
## Income.composition.of.resources       0.205988337 -0.254746613  0.44637038
## Schooling                             0.287997527 -0.225945962  0.43825108
## Developed                             0.307915267 -0.148590059  0.46668560
## Developing                           -0.307915267  0.148590059 -0.46668560
## Hepatitis.B_below90                  -0.086237946  0.197129865 -0.09988033
## Hepatitis.B_above90                   0.086237946 -0.197129865  0.09988033
## Polio_below90                        -0.155435726  0.270937717 -0.26859744
## Polio_above90                         0.155435726 -0.270937717  0.26859744
## Diphtheria_below90                   -0.176127349  0.263485357 -0.27212187
## Diphtheria_above90                    0.176127349 -0.263485357  0.27212187
##                                   Population thinness..1.19.years
## Life.expectancy                 -0.032942445           -0.4804714
## Adult.Mortality                 -0.001989446            0.3078186
## infant.deaths                    0.557267650            0.4646928
## Measles                          0.267155857            0.2234973
## BMI                             -0.069664918           -0.5356016
## under.five.deaths                0.544914569            0.4671530
## Total.expenditure               -0.057854707           -0.2908201
## HIV.AIDS                        -0.020012844            0.2017958
## GDP                             -0.031567137           -0.2817398
## Population                       1.000000000            0.2427012
## thinness..1.19.years             0.242701227            1.0000000
## thinness.5.9.years               0.240939542            0.9398249
## Income.composition.of.resources -0.020035719           -0.4378224
## Schooling                       -0.041171199           -0.4851424
## Developed                       -0.038613476           -0.3695766
## Developing                       0.038613476            0.3695766
## Hepatitis.B_below90              0.063201193            0.1721079
## Hepatitis.B_above90             -0.063201193           -0.1721079
## Polio_below90                    0.070275553            0.2731081
## Polio_above90                   -0.070275553           -0.2731081
## Diphtheria_below90               0.066990489            0.2800136
## Diphtheria_above90              -0.066990489           -0.2800136
##                                 thinness.5.9.years
## Life.expectancy                         -0.4743783
## Adult.Mortality                          0.3127501
## infant.deaths                            0.4703845
## Measles                                  0.2199085
## BMI                                     -0.5421874
## under.five.deaths                        0.4716814
## Total.expenditure                       -0.2979707
## HIV.AIDS                                 0.2047984
## GDP                                     -0.2853486
## Population                               0.2409395
## thinness..1.19.years                     0.9398249
## thinness.5.9.years                       1.0000000
## Income.composition.of.resources         -0.4267865
## Schooling                               -0.4730321
## Developed                               -0.3678503
## Developing                               0.3678503
## Hepatitis.B_below90                      0.1822763
## Hepatitis.B_above90                     -0.1822763
## Polio_below90                            0.2863211
## Polio_above90                           -0.2863211
## Diphtheria_below90                       0.2946470
## Diphtheria_above90                      -0.2946470
##                                 Income.composition.of.resources  Schooling
## Life.expectancy                                      0.73649241  0.7620802
## Adult.Mortality                                     -0.47491252 -0.4723905
## infant.deaths                                       -0.16470019 -0.2127197
## Measles                                             -0.15351757 -0.1652645
## BMI                                                  0.52084467  0.5606608
## under.five.deaths                                   -0.18385354 -0.2295452
## Total.expenditure                                    0.20598834  0.2879975
## HIV.AIDS                                            -0.25474661 -0.2259460
## GDP                                                  0.44637038  0.4382511
## Population                                          -0.02003572 -0.0411712
## thinness..1.19.years                                -0.43782239 -0.4851424
## thinness.5.9.years                                  -0.42678654 -0.4730321
## Income.composition.of.resources                      1.00000000  0.8108359
## Schooling                                            0.81083588  1.0000000
## Developed                                            0.48691956  0.5257973
## Developing                                          -0.48691956 -0.5257973
## Hepatitis.B_below90                                 -0.32977993 -0.3600401
## Hepatitis.B_above90                                  0.32977993  0.3600401
## Polio_below90                                       -0.47586703 -0.5182907
## Polio_above90                                        0.47586703  0.5182907
## Diphtheria_below90                                  -0.48667960 -0.5281486
## Diphtheria_above90                                   0.48667960  0.5281486
##                                   Developed  Developing Hepatitis.B_below90
## Life.expectancy                  0.48058927 -0.48058927         -0.39549144
## Adult.Mortality                 -0.31368202  0.31368202          0.26210909
## infant.deaths                   -0.11225158  0.11225158          0.17495190
## Measles                         -0.07695507  0.07695507          0.12264181
## BMI                              0.31377031 -0.31377031         -0.23793027
## under.five.deaths               -0.11519509  0.11519509          0.18566031
## Total.expenditure                0.30791527 -0.30791527         -0.08623795
## HIV.AIDS                        -0.14859006  0.14859006          0.19712986
## GDP                              0.46668560 -0.46668560         -0.09988033
## Population                      -0.03861348  0.03861348          0.06320119
## thinness..1.19.years            -0.36957657  0.36957657          0.17210792
## thinness.5.9.years              -0.36785032  0.36785032          0.18227633
## Income.composition.of.resources  0.48691956 -0.48691956         -0.32977993
## Schooling                        0.52579729 -0.52579729         -0.36004008
## Developed                        1.00000000 -1.00000000         -0.13660106
## Developing                      -1.00000000  1.00000000          0.13660106
## Hepatitis.B_below90             -0.13660106  0.13660106          1.00000000
## Hepatitis.B_above90              0.13660106 -0.13660106         -1.00000000
## Polio_below90                   -0.30401930  0.30401930          0.75920466
## Polio_above90                    0.30401930 -0.30401930         -0.75920466
## Diphtheria_below90              -0.31344853  0.31344853          0.78541073
## Diphtheria_above90               0.31344853 -0.31344853         -0.78541073
##                                 Hepatitis.B_above90 Polio_below90 Polio_above90
## Life.expectancy                          0.39549144   -0.56642658    0.56642658
## Adult.Mortality                         -0.26210909    0.36653693   -0.36653693
## infant.deaths                           -0.17495190    0.21411536   -0.21411536
## Measles                                 -0.12264181    0.15136310   -0.15136310
## BMI                                      0.23793027   -0.33780111    0.33780111
## under.five.deaths                       -0.18566031    0.22620022   -0.22620022
## Total.expenditure                        0.08623795   -0.15543573    0.15543573
## HIV.AIDS                                -0.19712986    0.27093772   -0.27093772
## GDP                                      0.09988033   -0.26859744    0.26859744
## Population                              -0.06320119    0.07027555   -0.07027555
## thinness..1.19.years                    -0.17210792    0.27310810   -0.27310810
## thinness.5.9.years                      -0.18227633    0.28632107   -0.28632107
## Income.composition.of.resources          0.32977993   -0.47586703    0.47586703
## Schooling                                0.36004008   -0.51829067    0.51829067
## Developed                                0.13660106   -0.30401930    0.30401930
## Developing                              -0.13660106    0.30401930   -0.30401930
## Hepatitis.B_below90                     -1.00000000    0.75920466   -0.75920466
## Hepatitis.B_above90                      1.00000000   -0.75920466    0.75920466
## Polio_below90                           -0.75920466    1.00000000   -1.00000000
## Polio_above90                            0.75920466   -1.00000000    1.00000000
## Diphtheria_below90                      -0.78541073    0.91241293   -0.91241293
## Diphtheria_above90                       0.78541073   -0.91241293    0.91241293
##                                 Diphtheria_below90 Diphtheria_above90
## Life.expectancy                        -0.56637789         0.56637789
## Adult.Mortality                         0.36305654        -0.36305654
## infant.deaths                           0.21238743        -0.21238743
## Measles                                 0.14053796        -0.14053796
## BMI                                    -0.33316029         0.33316029
## under.five.deaths                       0.22474158        -0.22474158
## Total.expenditure                      -0.17612735         0.17612735
## HIV.AIDS                                0.26348536        -0.26348536
## GDP                                    -0.27212187         0.27212187
## Population                              0.06699049        -0.06699049
## thinness..1.19.years                    0.28001364        -0.28001364
## thinness.5.9.years                      0.29464704        -0.29464704
## Income.composition.of.resources        -0.48667960         0.48667960
## Schooling                              -0.52814865         0.52814865
## Developed                              -0.31344853         0.31344853
## Developing                              0.31344853        -0.31344853
## Hepatitis.B_below90                     0.78541073        -0.78541073
## Hepatitis.B_above90                    -0.78541073         0.78541073
## Polio_below90                           0.91241293        -0.91241293
## Polio_above90                          -0.91241293         0.91241293
## Diphtheria_below90                      1.00000000        -1.00000000
## Diphtheria_above90                     -1.00000000         1.00000000
library(corrplot)
## corrplot 0.92 loaded
corrplot(life_cor, method = "number", addCoef.col = "black",number.cex=0.50)

Since it is hard to interpret, plotting correlation using ggcorr.

library(GGally)
## Registered S3 method overwritten by 'GGally':
##   method from   
##   +.gg   ggplot2
## 
## Attaching package: 'GGally'
## The following object is masked from 'package:ggmosaic':
## 
##     happy
data_num <- life_exp %>% 
                            select_if(is.numeric)
ggcorr(data_num, 
       label = T, 
       label_size = 2,
       label_round = 2,
       hjust = 1,
       size = 3, 
       color = "royalblue",
       layout.exp = 5,
       low = "green3", 
       mid = "gray95", 
       high = "darkorange",
       name = "Correlation")

In the model analysis, we will see that the dependent variable, Life Expectancy, has a moderately strong positive correlation with the independent variables, Education, and Income Composition of Resources. However, there is a bad link between it and adult mortality. This is a true fact, as a high adult mortality rate will inevitably result in a poor life expectancy for all.

A very little link exists between life expectancy and both the population and the measles. The following analysis will be used to evaluate it further.

Additionally, the Corr Matrix reveals a very significant link between baby deaths and under-five fatalities. This high correlation suggests that there is multicollinearity between them. Since other factors appear to be more closely related to situations during the neonatal period, we will deselect under 5.deaths.

life_exp1 <- life_exp %>% 
  select(-under.five.deaths)
head(life_exp1)
##   Life.expectancy Adult.Mortality infant.deaths Measles  BMI Total.expenditure
## 1            65.0             263            62    1154 19.1              8.16
## 2            59.9             271            64     492 18.6              8.18
## 3            59.9             268            66     430 18.1              8.13
## 4            59.5             272            69    2787 17.6              8.52
## 5            59.2             275            71    3013 17.2              7.87
## 6            58.8             279            74    1989 16.7              9.20
##   HIV.AIDS       GDP Population thinness..1.19.years thinness.5.9.years
## 1      0.1 584.25921   33736494                 17.2               17.3
## 2      0.1 612.69651     327582                 17.5               17.5
## 3      0.1 631.74498   31731688                 17.7               17.7
## 4      0.1 669.95900    3696958                 17.9               18.0
## 5      0.1  63.53723    2978599                 18.2               18.2
## 6      0.1 553.32894    2883167                 18.4               18.4
##   Income.composition.of.resources Schooling Developed Developing
## 1                           0.479      10.1         0          1
## 2                           0.476      10.0         0          1
## 3                           0.470       9.9         0          1
## 4                           0.463       9.8         0          1
## 5                           0.454       9.5         0          1
## 6                           0.448       9.2         0          1
##   Hepatitis.B_below90 Hepatitis.B_above90 Polio_below90 Polio_above90
## 1                   1                   0             1             0
## 2                   1                   0             1             0
## 3                   1                   0             1             0
## 4                   1                   0             1             0
## 5                   1                   0             1             0
## 6                   1                   0             1             0
##   Diphtheria_below90 Diphtheria_above90
## 1                  1                  0
## 2                  1                  0
## 3                  1                  0
## 4                  1                  0
## 5                  1                  0
## 6                  1                  0

Model Building

As was said at the outset of this Analysis, we’ll use a few key variables to forecast life expectancy. The complete linear prediction model is presented here.

Model1 <- lm(formula = Life.expectancy ~ ., data = life_exp1)
summary(Model1)
## 
## Call:
## lm(formula = Life.expectancy ~ ., data = life_exp1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -21.2891  -2.2731   0.0099   2.3397  18.8170 
## 
## Coefficients: (4 not defined because of singularities)
##                                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                      5.817e+01  5.358e-01 108.566  < 2e-16 ***
## Adult.Mortality                 -1.786e-02  7.917e-04 -22.561  < 2e-16 ***
## infant.deaths                   -1.007e-03  9.242e-04  -1.090  0.27587    
## Measles                         -1.756e-05  7.524e-06  -2.334  0.01964 *  
## BMI                              3.709e-02  4.927e-03   7.528 6.80e-14 ***
## Total.expenditure                5.264e-02  3.254e-02   1.618  0.10581    
## HIV.AIDS                        -4.806e-01  1.741e-02 -27.608  < 2e-16 ***
## GDP                              4.715e-05  6.637e-06   7.104 1.52e-12 ***
## Population                       2.389e-09  1.638e-09   1.458  0.14488    
## thinness..1.19.years            -1.251e-01  4.973e-02  -2.515  0.01194 *  
## thinness.5.9.years               6.262e-02  4.902e-02   1.278  0.20150    
## Income.composition.of.resources  7.463e+00  6.185e-01  12.067  < 2e-16 ***
## Schooling                        7.797e-01  4.126e-02  18.898  < 2e-16 ***
## Developed                        6.764e-01  2.503e-01   2.703  0.00692 ** 
## Developing                              NA         NA      NA       NA    
## Hepatitis.B_below90              3.184e-01  2.497e-01   1.275  0.20246    
## Hepatitis.B_above90                     NA         NA      NA       NA    
## Polio_below90                   -1.515e+00  3.755e-01  -4.035 5.60e-05 ***
## Polio_above90                           NA         NA      NA       NA    
## Diphtheria_below90              -1.110e+00  3.997e-01  -2.776  0.00554 ** 
## Diphtheria_above90                      NA         NA      NA       NA    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.003 on 2921 degrees of freedom
## Multiple R-squared:  0.824,  Adjusted R-squared:  0.8231 
## F-statistic: 854.8 on 16 and 2921 DF,  p-value: < 2.2e-16

Advanced Feature Selection

We will now choose the most crucial predictors using R’s automatic calculation.

Backward Direction
model_backward <- stepAIC(Model1, direction = "backward")
## Start:  AIC=8166.65
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Hepatitis.B_above90 + 
##     Polio_below90 + Polio_above90 + Diphtheria_below90 + Diphtheria_above90
## 
## 
## Step:  AIC=8166.65
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Hepatitis.B_above90 + 
##     Polio_below90 + Polio_above90 + Diphtheria_below90
## 
## 
## Step:  AIC=8166.65
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Hepatitis.B_above90 + 
##     Polio_below90 + Diphtheria_below90
## 
## 
## Step:  AIC=8166.65
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Polio_below90 + 
##     Diphtheria_below90
## 
## 
## Step:  AIC=8166.65
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Hepatitis.B_below90 + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - infant.deaths                    1      19.0 46816 8165.8
## - Hepatitis.B_below90              1      26.0 46823 8166.3
## - thinness.5.9.years               1      26.1 46823 8166.3
## <none>                                         46797 8166.7
## - Population                       1      34.1 46831 8166.8
## - Total.expenditure                1      41.9 46839 8167.3
## - Measles                          1      87.3 46884 8170.1
## - thinness..1.19.years             1     101.4 46898 8171.0
## - Developed                        1     117.0 46914 8172.0
## - Diphtheria_below90               1     123.5 46921 8172.4
## - Polio_below90                    1     260.8 47058 8181.0
## - GDP                              1     808.5 47606 8215.0
## - BMI                              1     908.0 47705 8221.1
## - Income.composition.of.resources  1    2332.9 49130 8307.6
## - Schooling                        1    5721.7 52519 8503.6
## - Adult.Mortality                  1    8154.5 54952 8636.6
## - HIV.AIDS                         1   12211.4 59008 8845.9
## 
## Step:  AIC=8165.85
## Life.expectancy ~ Adult.Mortality + Measles + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + Population + thinness..1.19.years + thinness.5.9.years + 
##     Income.composition.of.resources + Schooling + Developed + 
##     Hepatitis.B_below90 + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - Population                       1      18.4 46835 8165.0
## - thinness.5.9.years               1      21.7 46838 8165.2
## - Hepatitis.B_below90              1      25.4 46841 8165.4
## <none>                                         46816 8165.8
## - Total.expenditure                1      40.6 46857 8166.4
## - thinness..1.19.years             1     107.0 46923 8170.6
## - Developed                        1     112.0 46928 8170.9
## - Diphtheria_below90               1     125.9 46942 8171.7
## - Measles                          1     148.5 46965 8173.2
## - Polio_below90                    1     264.2 47080 8180.4
## - GDP                              1     808.8 47625 8214.2
## - BMI                              1     900.2 47716 8219.8
## - Income.composition.of.resources  1    2317.7 49134 8305.8
## - Schooling                        1    5748.3 52564 8504.1
## - Adult.Mortality                  1    8147.9 54964 8635.3
## - HIV.AIDS                         1   12197.0 59013 8844.1
## 
## Step:  AIC=8165.01
## Life.expectancy ~ Adult.Mortality + Measles + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + thinness..1.19.years + thinness.5.9.years + 
##     Income.composition.of.resources + Schooling + Developed + 
##     Hepatitis.B_below90 + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - thinness.5.9.years               1      23.6 46858 8164.5
## - Hepatitis.B_below90              1      26.2 46861 8164.6
## <none>                                         46835 8165.0
## - Total.expenditure                1      41.2 46876 8165.6
## - thinness..1.19.years             1     102.1 46937 8169.4
## - Developed                        1     112.7 46947 8170.1
## - Diphtheria_below90               1     125.9 46960 8170.9
## - Measles                          1     132.5 46967 8171.3
## - Polio_below90                    1     262.3 47097 8179.4
## - GDP                              1     807.1 47642 8213.2
## - BMI                              1     914.3 47749 8219.8
## - Income.composition.of.resources  1    2341.1 49176 8306.3
## - Schooling                        1    5771.0 52606 8504.4
## - Adult.Mortality                  1    8153.1 54988 8634.5
## - HIV.AIDS                         1   12267.7 59102 8846.5
## 
## Step:  AIC=8164.48
## Life.expectancy ~ Adult.Mortality + Measles + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + thinness..1.19.years + Income.composition.of.resources + 
##     Schooling + Developed + Hepatitis.B_below90 + Polio_below90 + 
##     Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - Hepatitis.B_below90              1      25.5 46884 8164.1
## <none>                                         46858 8164.5
## - Total.expenditure                1      38.2 46896 8164.9
## - Developed                        1     110.2 46968 8169.4
## - Diphtheria_below90               1     122.4 46980 8170.1
## - Measles                          1     130.3 46988 8170.6
## - thinness..1.19.years             1     178.7 47037 8173.7
## - Polio_below90                    1     261.8 47120 8178.9
## - GDP                              1     800.3 47658 8212.2
## - BMI                              1     893.0 47751 8217.9
## - Income.composition.of.resources  1    2351.6 49210 8306.3
## - Schooling                        1    5799.4 52657 8505.3
## - Adult.Mortality                  1    8137.6 54996 8632.9
## - HIV.AIDS                         1   12261.7 59120 8845.4
## 
## Step:  AIC=8164.08
## Life.expectancy ~ Adult.Mortality + Measles + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + thinness..1.19.years + Income.composition.of.resources + 
##     Schooling + Developed + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## <none>                                         46884 8164.1
## - Total.expenditure                1      40.5 46924 8164.6
## - Diphtheria_below90               1      97.8 46981 8168.2
## - Developed                        1     122.5 47006 8169.7
## - Measles                          1     127.4 47011 8170.1
## - thinness..1.19.years             1     181.1 47065 8173.4
## - Polio_below90                    1     240.8 47124 8177.1
## - GDP                              1     853.0 47737 8215.1
## - BMI                              1     886.9 47771 8217.1
## - Income.composition.of.resources  1    2352.2 49236 8305.9
## - Schooling                        1    5809.8 52693 8505.3
## - Adult.Mortality                  1    8124.3 55008 8631.6
## - HIV.AIDS                         1   12291.1 59175 8846.1
summary(model_backward)
## 
## Call:
## lm(formula = Life.expectancy ~ Adult.Mortality + Measles + BMI + 
##     Total.expenditure + HIV.AIDS + GDP + thinness..1.19.years + 
##     Income.composition.of.resources + Schooling + Developed + 
##     Polio_below90 + Diphtheria_below90, data = life_exp1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -21.2357  -2.2519  -0.0019   2.3290  18.8733 
## 
## Coefficients:
##                                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                      5.821e+01  5.296e-01 109.932  < 2e-16 ***
## Adult.Mortality                 -1.782e-02  7.916e-04 -22.514  < 2e-16 ***
## Measles                         -1.885e-05  6.686e-06  -2.820 0.004839 ** 
## BMI                              3.636e-02  4.888e-03   7.439 1.33e-13 ***
## Total.expenditure                5.161e-02  3.247e-02   1.590 0.112032    
## HIV.AIDS                        -4.807e-01  1.736e-02 -27.692  < 2e-16 ***
## GDP                              4.799e-05  6.579e-06   7.295 3.82e-13 ***
## thinness..1.19.years            -7.126e-02  2.120e-02  -3.362 0.000785 ***
## Income.composition.of.resources  7.475e+00  6.171e-01  12.114  < 2e-16 ***
## Schooling                        7.846e-01  4.121e-02  19.038  < 2e-16 ***
## Developed                        6.870e-01  2.485e-01   2.765 0.005736 ** 
## Polio_below90                   -1.431e+00  3.691e-01  -3.876 0.000108 ***
## Diphtheria_below90              -9.148e-01  3.703e-01  -2.470 0.013560 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.004 on 2925 degrees of freedom
## Multiple R-squared:  0.8237, Adjusted R-squared:  0.823 
## F-statistic:  1139 on 12 and 2925 DF,  p-value: < 2.2e-16
Forward Direction
Model2 <- lm(Life.expectancy ~ 1, data = life_exp1)
model_forward <- stepAIC(Model2, scope = list(lower = Model2, upper = Model1), direction = "forward")
## Start:  AIC=13239.02
## Life.expectancy ~ 1
## 
##                                   Df Sum of Sq    RSS   AIC
## + Schooling                        1    154435 111481 10687
## + Income.composition.of.resources  1    144239 121678 10944
## + Adult.Mortality                  1    129046 136870 11290
## + BMI                              1     86804 179113 12080
## + Polio_below90                    1     85316 180600 12104
## + Polio_above90                    1     85316 180600 12104
## + Diphtheria_below90               1     85302 180615 12105
## + Diphtheria_above90               1     85302 180615 12105
## + HIV.AIDS                         1     82413 183503 12151
## + Developed                        1     61418 204499 12469
## + Developing                       1     61418 204499 12469
## + thinness..1.19.years             1     61388 204529 12470
## + thinness.5.9.years               1     59840 206076 12492
## + GDP                              1     53651 212265 12579
## + Hepatitis.B_below90              1     41593 224324 12741
## + Hepatitis.B_above90              1     41593 224324 12741
## + Total.expenditure                1     14122 251795 13081
## + infant.deaths                    1     10313 255604 13125
## + Measles                          1      6630 259287 13167
## + Population                       1       289 265628 13238
## <none>                                         265917 13239
## 
## Step:  AIC=10686.94
## Life.expectancy ~ Schooling
## 
##                                   Df Sum of Sq    RSS     AIC
## + HIV.AIDS                         1     41432  70049  9323.8
## + Adult.Mortality                  1     38789  72692  9432.6
## + Income.composition.of.resources  1     10914 100567 10386.2
## + Polio_below90                    1     10687 100794 10392.9
## + Polio_above90                    1     10687 100794 10392.9
## + Diphtheria_below90               1      9905 101576 10415.6
## + Diphtheria_above90               1      9905 101576 10415.6
## + BMI                              1      8050 103431 10468.7
## + Hepatitis.B_below90              1      4481 107000 10568.4
## + Hepatitis.B_above90              1      4481 107000 10568.4
## + thinness.5.9.years               1      4443 107038 10569.4
## + GDP                              1      4367 107114 10571.5
## + thinness..1.19.years             1      4266 107215 10574.3
## + Developed                        1      2346 109136 10626.5
## + Developing                       1      2346 109136 10626.5
## + infant.deaths                    1       338 111144 10680.0
## + Measles                          1       279 111202 10681.6
## <none>                                         111481 10686.9
## + Total.expenditure                1        35 111446 10688.0
## + Population                       1         1 111481 10688.9
## 
## Step:  AIC=9323.78
## Life.expectancy ~ Schooling + HIV.AIDS
## 
##                                   Df Sum of Sq   RSS    AIC
## + Adult.Mortality                  1   12581.1 57468 8744.2
## + Income.composition.of.resources  1    6329.3 63720 9047.6
## + Polio_below90                    1    4481.6 65568 9131.5
## + Polio_above90                    1    4481.6 65568 9131.5
## + Diphtheria_below90               1    4231.3 65818 9142.7
## + Diphtheria_above90               1    4231.3 65818 9142.7
## + BMI                              1    3738.7 66311 9164.6
## + GDP                              1    3537.1 66512 9173.6
## + thinness.5.9.years               1    1911.6 68138 9244.5
## + thinness..1.19.years             1    1895.9 68153 9245.2
## + Hepatitis.B_below90              1    1709.7 68340 9253.2
## + Hepatitis.B_above90              1    1709.7 68340 9253.2
## + Developed                        1    1692.6 68357 9253.9
## + Developing                       1    1692.6 68357 9253.9
## + infant.deaths                    1     541.3 69508 9303.0
## + Total.expenditure                1     358.9 69691 9310.7
## + Measles                          1     326.7 69723 9312.1
## + Population                       1      48.2 70001 9323.8
## <none>                                         70049 9323.8
## 
## Step:  AIC=8744.16
## Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality
## 
##                                   Df Sum of Sq   RSS    AIC
## + Income.composition.of.resources  1    4225.0 53243 8521.8
## + Polio_below90                    1    3338.7 54130 8570.3
## + Polio_above90                    1    3338.7 54130 8570.3
## + Diphtheria_below90               1    3190.2 54278 8578.4
## + Diphtheria_above90               1    3190.2 54278 8578.4
## + BMI                              1    2281.6 55187 8627.1
## + GDP                              1    2256.1 55212 8628.5
## + thinness..1.19.years             1    1385.3 56083 8674.5
## + thinness.5.9.years               1    1304.3 56164 8678.7
## + Hepatitis.B_below90              1    1220.5 56248 8683.1
## + Hepatitis.B_above90              1    1220.5 56248 8683.1
## + Developed                        1    1043.4 56425 8692.3
## + Developing                       1    1043.4 56425 8692.3
## + infant.deaths                    1     621.4 56847 8714.2
## + Measles                          1     602.5 56866 8715.2
## + Total.expenditure                1     262.3 57206 8732.7
## + Population                       1      67.1 57401 8742.7
## <none>                                         57468 8744.2
## 
## Step:  AIC=8521.81
## Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality + Income.composition.of.resources
## 
##                        Df Sum of Sq   RSS    AIC
## + Polio_below90         1   2783.03 50460 8366.1
## + Polio_above90         1   2783.03 50460 8366.1
## + Diphtheria_below90    1   2593.54 50650 8377.1
## + Diphtheria_above90    1   2593.54 50650 8377.1
## + BMI                   1   1689.43 51554 8429.1
## + GDP                   1   1421.53 51822 8444.3
## + thinness..1.19.years  1   1083.54 52160 8463.4
## + thinness.5.9.years    1   1033.44 52210 8466.2
## + Hepatitis.B_below90   1   1019.04 52224 8467.0
## + Hepatitis.B_above90   1   1019.04 52224 8467.0
## + infant.deaths         1    650.19 52593 8487.7
## + Developed             1    644.29 52599 8488.0
## + Developing            1    644.29 52599 8488.0
## + Measles               1    473.60 52770 8497.6
## + Total.expenditure     1    367.06 52876 8503.5
## + Population            1     87.46 53156 8519.0
## <none>                              53243 8521.8
## 
## Step:  AIC=8366.08
## Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality + Income.composition.of.resources + 
##     Polio_below90
## 
##                        Df Sum of Sq   RSS    AIC
## + BMI                   1   1591.39 48869 8273.9
## + GDP                   1   1314.00 49146 8290.6
## + thinness..1.19.years  1   1086.74 49374 8304.1
## + thinness.5.9.years    1    954.28 49506 8312.0
## + Developed             1    586.01 49874 8333.8
## + Developing            1    586.01 49874 8333.8
## + infant.deaths         1    346.13 50114 8347.9
## + Total.expenditure     1    325.16 50135 8349.1
## + Measles               1    304.92 50155 8350.3
## + Diphtheria_below90    1     98.56 50362 8362.3
## + Diphtheria_above90    1     98.56 50362 8362.3
## + Hepatitis.B_below90   1     62.07 50398 8364.5
## + Hepatitis.B_above90   1     62.07 50398 8364.5
## <none>                              50460 8366.1
## + Population            1     34.14 50426 8366.1
## 
## Step:  AIC=8273.93
## Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality + Income.composition.of.resources + 
##     Polio_below90 + BMI
## 
##                        Df Sum of Sq   RSS    AIC
## + GDP                   1   1225.94 47643 8201.3
## + Developing            1    585.34 48284 8240.5
## + Developed             1    585.34 48284 8240.5
## + thinness..1.19.years  1    416.90 48452 8250.8
## + thinness.5.9.years    1    313.32 48556 8257.0
## + Measles               1    178.58 48690 8265.2
## + infant.deaths         1    171.23 48698 8265.6
## + Total.expenditure     1    166.56 48702 8265.9
## + Diphtheria_below90    1    116.41 48753 8268.9
## + Diphtheria_above90    1    116.41 48753 8268.9
## + Hepatitis.B_below90   1     62.77 48806 8272.2
## + Hepatitis.B_above90   1     62.77 48806 8272.2
## <none>                              48869 8273.9
## + Population            1     10.86 48858 8275.3
## 
## Step:  AIC=8201.29
## Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality + Income.composition.of.resources + 
##     Polio_below90 + BMI + GDP
## 
##                        Df Sum of Sq   RSS    AIC
## + thinness..1.19.years  1    333.22 47310 8182.7
## + thinness.5.9.years    1    230.47 47413 8189.0
## + Developed             1    215.12 47428 8190.0
## + Developing            1    215.12 47428 8190.0
## + Measles               1    182.92 47460 8192.0
## + infant.deaths         1    161.07 47482 8193.3
## + Total.expenditure     1    148.67 47494 8194.1
## + Diphtheria_below90    1    112.31 47531 8196.4
## + Diphtheria_above90    1    112.31 47531 8196.4
## <none>                              47643 8201.3
## + Population            1      7.35 47636 8202.8
## + Hepatitis.B_below90   1      6.34 47637 8202.9
## + Hepatitis.B_above90   1      6.34 47637 8202.9
## 
## Step:  AIC=8182.67
## Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality + Income.composition.of.resources + 
##     Polio_below90 + BMI + GDP + thinness..1.19.years
## 
##                       Df Sum of Sq   RSS    AIC
## + Developed            1   150.343 47159 8175.3
## + Developing           1   150.343 47159 8175.3
## + Measles              1   121.241 47189 8177.1
## + Diphtheria_below90   1   106.141 47204 8178.1
## + Diphtheria_above90   1   106.141 47204 8178.1
## + Total.expenditure    1    89.467 47220 8179.1
## <none>                             47310 8182.7
## + infant.deaths        1    31.366 47278 8182.7
## + thinness.5.9.years   1    11.573 47298 8183.9
## + Population           1     4.207 47306 8184.4
## + Hepatitis.B_below90  1     4.175 47306 8184.4
## + Hepatitis.B_above90  1     4.175 47306 8184.4
## 
## Step:  AIC=8175.31
## Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality + Income.composition.of.resources + 
##     Polio_below90 + BMI + GDP + thinness..1.19.years + Developed
## 
##                       Df Sum of Sq   RSS    AIC
## + Measles              1   130.651 47029 8169.2
## + Diphtheria_below90   1   102.239 47057 8170.9
## + Diphtheria_above90   1   102.239 47057 8170.9
## + Total.expenditure    1    53.267 47106 8174.0
## + infant.deaths        1    41.453 47118 8174.7
## <none>                             47159 8175.3
## + thinness.5.9.years   1    14.863 47145 8176.4
## + Population           1     3.368 47156 8177.1
## + Hepatitis.B_below90  1     0.730 47159 8177.3
## + Hepatitis.B_above90  1     0.730 47159 8177.3
## 
## Step:  AIC=8169.16
## Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality + Income.composition.of.resources + 
##     Polio_below90 + BMI + GDP + thinness..1.19.years + Developed + 
##     Measles
## 
##                       Df Sum of Sq   RSS    AIC
## + Diphtheria_below90   1   104.656 46924 8164.6
## + Diphtheria_above90   1   104.656 46924 8164.6
## + Total.expenditure    1    47.352 46981 8168.2
## <none>                             47029 8169.2
## + Population           1    20.908 47008 8169.9
## + thinness.5.9.years   1    16.771 47012 8170.1
## + infant.deaths        1     1.871 47027 8171.0
## + Hepatitis.B_below90  1     1.145 47028 8171.1
## + Hepatitis.B_above90  1     1.145 47028 8171.1
## 
## Step:  AIC=8164.62
## Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality + Income.composition.of.resources + 
##     Polio_below90 + BMI + GDP + thinness..1.19.years + Developed + 
##     Measles + Diphtheria_below90
## 
##                       Df Sum of Sq   RSS    AIC
## + Total.expenditure    1    40.502 46884 8164.1
## <none>                             46924 8164.6
## + Hepatitis.B_below90  1    27.810 46896 8164.9
## + Hepatitis.B_above90  1    27.810 46896 8164.9
## + Population           1    21.718 46902 8165.3
## + thinness.5.9.years   1    19.742 46904 8165.4
## + infant.deaths        1     1.029 46923 8166.6
## 
## Step:  AIC=8164.08
## Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality + Income.composition.of.resources + 
##     Polio_below90 + BMI + GDP + thinness..1.19.years + Developed + 
##     Measles + Diphtheria_below90 + Total.expenditure
## 
##                       Df Sum of Sq   RSS    AIC
## <none>                             46884 8164.1
## + Hepatitis.B_below90  1   25.4928 46858 8164.5
## + Hepatitis.B_above90  1   25.4928 46858 8164.5
## + thinness.5.9.years   1   22.8700 46861 8164.6
## + Population           1   21.1256 46862 8164.8
## + infant.deaths        1    1.3254 46882 8166.0
summary(model_forward)
## 
## Call:
## lm(formula = Life.expectancy ~ Schooling + HIV.AIDS + Adult.Mortality + 
##     Income.composition.of.resources + Polio_below90 + BMI + GDP + 
##     thinness..1.19.years + Developed + Measles + Diphtheria_below90 + 
##     Total.expenditure, data = life_exp1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -21.2357  -2.2519  -0.0019   2.3290  18.8733 
## 
## Coefficients:
##                                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                      5.821e+01  5.296e-01 109.932  < 2e-16 ***
## Schooling                        7.846e-01  4.121e-02  19.038  < 2e-16 ***
## HIV.AIDS                        -4.807e-01  1.736e-02 -27.692  < 2e-16 ***
## Adult.Mortality                 -1.782e-02  7.916e-04 -22.514  < 2e-16 ***
## Income.composition.of.resources  7.475e+00  6.171e-01  12.114  < 2e-16 ***
## Polio_below90                   -1.431e+00  3.691e-01  -3.876 0.000108 ***
## BMI                              3.636e-02  4.888e-03   7.439 1.33e-13 ***
## GDP                              4.799e-05  6.579e-06   7.295 3.82e-13 ***
## thinness..1.19.years            -7.126e-02  2.120e-02  -3.362 0.000785 ***
## Developed                        6.870e-01  2.485e-01   2.765 0.005736 ** 
## Measles                         -1.885e-05  6.686e-06  -2.820 0.004839 ** 
## Diphtheria_below90              -9.148e-01  3.703e-01  -2.470 0.013560 *  
## Total.expenditure                5.161e-02  3.247e-02   1.590 0.112032    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.004 on 2925 degrees of freedom
## Multiple R-squared:  0.8237, Adjusted R-squared:  0.823 
## F-statistic:  1139 on 12 and 2925 DF,  p-value: < 2.2e-16
Both Direction
model_both <- stepAIC(Model1, scope = list(lower = Model2, upper = Model1), direction = "both")
## Start:  AIC=8166.65
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Hepatitis.B_above90 + 
##     Polio_below90 + Polio_above90 + Diphtheria_below90 + Diphtheria_above90
## 
## 
## Step:  AIC=8166.65
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Hepatitis.B_above90 + 
##     Polio_below90 + Polio_above90 + Diphtheria_below90
## 
## 
## Step:  AIC=8166.65
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Hepatitis.B_above90 + 
##     Polio_below90 + Diphtheria_below90
## 
## 
## Step:  AIC=8166.65
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Polio_below90 + 
##     Diphtheria_below90
## 
## 
## Step:  AIC=8166.65
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Hepatitis.B_below90 + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - infant.deaths                    1      19.0 46816 8165.8
## - Hepatitis.B_below90              1      26.0 46823 8166.3
## - thinness.5.9.years               1      26.1 46823 8166.3
## <none>                                         46797 8166.7
## - Population                       1      34.1 46831 8166.8
## - Total.expenditure                1      41.9 46839 8167.3
## - Measles                          1      87.3 46884 8170.1
## - thinness..1.19.years             1     101.4 46898 8171.0
## - Developed                        1     117.0 46914 8172.0
## - Diphtheria_below90               1     123.5 46921 8172.4
## - Polio_below90                    1     260.8 47058 8181.0
## - GDP                              1     808.5 47606 8215.0
## - BMI                              1     908.0 47705 8221.1
## - Income.composition.of.resources  1    2332.9 49130 8307.6
## - Schooling                        1    5721.7 52519 8503.6
## - Adult.Mortality                  1    8154.5 54952 8636.6
## - HIV.AIDS                         1   12211.4 59008 8845.9
## 
## Step:  AIC=8165.85
## Life.expectancy ~ Adult.Mortality + Measles + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + Population + thinness..1.19.years + thinness.5.9.years + 
##     Income.composition.of.resources + Schooling + Developed + 
##     Hepatitis.B_below90 + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - Population                       1      18.4 46835 8165.0
## - thinness.5.9.years               1      21.7 46838 8165.2
## - Hepatitis.B_below90              1      25.4 46841 8165.4
## <none>                                         46816 8165.8
## - Total.expenditure                1      40.6 46857 8166.4
## + infant.deaths                    1      19.0 46797 8166.7
## - thinness..1.19.years             1     107.0 46923 8170.6
## - Developed                        1     112.0 46928 8170.9
## - Diphtheria_below90               1     125.9 46942 8171.7
## - Measles                          1     148.5 46965 8173.2
## - Polio_below90                    1     264.2 47080 8180.4
## - GDP                              1     808.8 47625 8214.2
## - BMI                              1     900.2 47716 8219.8
## - Income.composition.of.resources  1    2317.7 49134 8305.8
## - Schooling                        1    5748.3 52564 8504.1
## - Adult.Mortality                  1    8147.9 54964 8635.3
## - HIV.AIDS                         1   12197.0 59013 8844.1
## 
## Step:  AIC=8165.01
## Life.expectancy ~ Adult.Mortality + Measles + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + thinness..1.19.years + thinness.5.9.years + 
##     Income.composition.of.resources + Schooling + Developed + 
##     Hepatitis.B_below90 + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - thinness.5.9.years               1      23.6 46858 8164.5
## - Hepatitis.B_below90              1      26.2 46861 8164.6
## <none>                                         46835 8165.0
## - Total.expenditure                1      41.2 46876 8165.6
## + Population                       1      18.4 46816 8165.8
## + infant.deaths                    1       3.4 46831 8166.8
## - thinness..1.19.years             1     102.1 46937 8169.4
## - Developed                        1     112.7 46947 8170.1
## - Diphtheria_below90               1     125.9 46960 8170.9
## - Measles                          1     132.5 46967 8171.3
## - Polio_below90                    1     262.3 47097 8179.4
## - GDP                              1     807.1 47642 8213.2
## - BMI                              1     914.3 47749 8219.8
## - Income.composition.of.resources  1    2341.1 49176 8306.3
## - Schooling                        1    5771.0 52606 8504.4
## - Adult.Mortality                  1    8153.1 54988 8634.5
## - HIV.AIDS                         1   12267.7 59102 8846.5
## 
## Step:  AIC=8164.48
## Life.expectancy ~ Adult.Mortality + Measles + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + thinness..1.19.years + Income.composition.of.resources + 
##     Schooling + Developed + Hepatitis.B_below90 + Polio_below90 + 
##     Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - Hepatitis.B_below90              1      25.5 46884 8164.1
## <none>                                         46858 8164.5
## - Total.expenditure                1      38.2 46896 8164.9
## + thinness.5.9.years               1      23.6 46835 8165.0
## + Population                       1      20.3 46838 8165.2
## + infant.deaths                    1       1.6 46857 8166.4
## - Developed                        1     110.2 46968 8169.4
## - Diphtheria_below90               1     122.4 46980 8170.1
## - Measles                          1     130.3 46988 8170.6
## - thinness..1.19.years             1     178.7 47037 8173.7
## - Polio_below90                    1     261.8 47120 8178.9
## - GDP                              1     800.3 47658 8212.2
## - BMI                              1     893.0 47751 8217.9
## - Income.composition.of.resources  1    2351.6 49210 8306.3
## - Schooling                        1    5799.4 52657 8505.3
## - Adult.Mortality                  1    8137.6 54996 8632.9
## - HIV.AIDS                         1   12261.7 59120 8845.4
## 
## Step:  AIC=8164.08
## Life.expectancy ~ Adult.Mortality + Measles + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + thinness..1.19.years + Income.composition.of.resources + 
##     Schooling + Developed + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## <none>                                         46884 8164.1
## + Hepatitis.B_below90              1      25.5 46858 8164.5
## + Hepatitis.B_above90              1      25.5 46858 8164.5
## - Total.expenditure                1      40.5 46924 8164.6
## + thinness.5.9.years               1      22.9 46861 8164.6
## + Population                       1      21.1 46862 8164.8
## + infant.deaths                    1       1.3 46882 8166.0
## - Diphtheria_below90               1      97.8 46981 8168.2
## - Developed                        1     122.5 47006 8169.7
## - Measles                          1     127.4 47011 8170.1
## - thinness..1.19.years             1     181.1 47065 8173.4
## - Polio_below90                    1     240.8 47124 8177.1
## - GDP                              1     853.0 47737 8215.1
## - BMI                              1     886.9 47771 8217.1
## - Income.composition.of.resources  1    2352.2 49236 8305.9
## - Schooling                        1    5809.8 52693 8505.3
## - Adult.Mortality                  1    8124.3 55008 8631.6
## - HIV.AIDS                         1   12291.1 59175 8846.1
summary(model_both)
## 
## Call:
## lm(formula = Life.expectancy ~ Adult.Mortality + Measles + BMI + 
##     Total.expenditure + HIV.AIDS + GDP + thinness..1.19.years + 
##     Income.composition.of.resources + Schooling + Developed + 
##     Polio_below90 + Diphtheria_below90, data = life_exp1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -21.2357  -2.2519  -0.0019   2.3290  18.8733 
## 
## Coefficients:
##                                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                      5.821e+01  5.296e-01 109.932  < 2e-16 ***
## Adult.Mortality                 -1.782e-02  7.916e-04 -22.514  < 2e-16 ***
## Measles                         -1.885e-05  6.686e-06  -2.820 0.004839 ** 
## BMI                              3.636e-02  4.888e-03   7.439 1.33e-13 ***
## Total.expenditure                5.161e-02  3.247e-02   1.590 0.112032    
## HIV.AIDS                        -4.807e-01  1.736e-02 -27.692  < 2e-16 ***
## GDP                              4.799e-05  6.579e-06   7.295 3.82e-13 ***
## thinness..1.19.years            -7.126e-02  2.120e-02  -3.362 0.000785 ***
## Income.composition.of.resources  7.475e+00  6.171e-01  12.114  < 2e-16 ***
## Schooling                        7.846e-01  4.121e-02  19.038  < 2e-16 ***
## Developed                        6.870e-01  2.485e-01   2.765 0.005736 ** 
## Polio_below90                   -1.431e+00  3.691e-01  -3.876 0.000108 ***
## Diphtheria_below90              -9.148e-01  3.703e-01  -2.470 0.013560 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.004 on 2925 degrees of freedom
## Multiple R-squared:  0.8237, Adjusted R-squared:  0.823 
## F-statistic:  1139 on 12 and 2925 DF,  p-value: < 2.2e-16
All possible (Regsubsets)
library(leaps)
regs <- regsubsets(Life.expectancy ~., data = life_exp1, nbest=10)
## Warning in leaps.setup(x, y, wt = wt, nbest = nbest, nvmax = nvmax, force.in =
## force.in, : 4 linear dependencies found
## Reordering variables and trying again:
plot(regs, 
     scale="adjr", 
     main="All possible regression: ranked by Adjusted R-squared")

The following variables, based on the supplied plot, are the most important: Adult Mortality, BMI, GDP, HIV AIDS, thinness..1.19.years, Income.composition.of.Resources, and Schooling. The significance of the p-value for the specified variables was also reflected in other models (three stars/***).

Model Construction Using Selected Variables
model_regs <- lm(formula = Life.expectancy ~ Adult.Mortality + BMI + GDP + HIV.AIDS + thinness..1.19.years + Income.composition.of.resources + Schooling, data = life_exp1)
summary(model_regs)
## 
## Call:
## lm(formula = Life.expectancy ~ Adult.Mortality + BMI + GDP + 
##     HIV.AIDS + thinness..1.19.years + Income.composition.of.resources + 
##     Schooling, data = life_exp1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -21.3454  -2.1298  -0.0851   2.2554  21.8040 
## 
## Coefficients:
##                                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                      5.545e+01  4.629e-01 119.805  < 2e-16 ***
## Adult.Mortality                 -1.843e-02  8.110e-04 -22.727  < 2e-16 ***
## BMI                              3.805e-02  5.006e-03   7.600 3.96e-14 ***
## GDP                              5.542e-05  6.493e-06   8.535  < 2e-16 ***
## HIV.AIDS                        -5.047e-01  1.770e-02 -28.512  < 2e-16 ***
## thinness..1.19.years            -9.097e-02  2.124e-02  -4.284 1.90e-05 ***
## Income.composition.of.resources  8.178e+00  6.305e-01  12.972  < 2e-16 ***
## Schooling                        9.373e-01  4.043e-02  23.184  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.127 on 2930 degrees of freedom
## Multiple R-squared:  0.8123, Adjusted R-squared:  0.8118 
## F-statistic:  1811 on 7 and 2930 DF,  p-value: < 2.2e-16

Compare the Adj. R-Squared from All Models

data.frame(model = c("model_backward","model_forward","model_both", "model_regs"), 
           AdjRsquare = c(summary(model_backward)$adj.r.square,
                          summary(model_forward)$adj.r.square,
                          summary(model_both)$adj.r.square,
                          summary(model_regs)$adj.r.square))
##            model AdjRsquare
## 1 model_backward  0.8229672
## 2  model_forward  0.8229672
## 3     model_both  0.8229672
## 4     model_regs  0.8118388

The model we will use to predict the ‘Life.expectancy’ will be model_backward, which we will select based on the supplied result.

Prediction

newData = data.frame(Adult.Mortality=55, Measles=4, BMI=23, Total.expenditure=1.23, HIV.AIDS = 23, GDP = 4235, thinness..1.19.years = 12, Income.composition.of.resources = 0.8, Schooling = 19, Developed = 0, Polio_below90 = 0, Diphtheria_below90 = 1)
predict(model_backward, newData)
##        1 
## 66.39876
predict(model_backward, newData, interval="confidence", level = 0.99)
##        fit     lwr      upr
## 1 66.39876 64.7485 68.04902
predicted=predict(model_backward, life_exp1)
predicted
##        1        2        3        4        5        6        7        8 
## 62.53524 62.26711 62.16433 61.90735 61.44841 61.17593 60.75759 60.41790 
##        9       10       11       12       13       14       15       16 
## 59.79887 59.47177 59.36032 58.35034 57.98504 62.59840 57.91175 57.52762 
##       17       18       19       20       21       22       23       24 
## 76.20788 77.37115 75.95013 75.82490 74.96086 73.94551 73.84116 75.01618 
##       25       26       27       28       29       30       31       32 
## 74.46019 70.94189 73.50867 73.60120 73.21808 73.26021 73.07832 73.12305 
##       33       34       35       36       37       38       39       40 
## 76.84333 76.90202 75.26686 75.12568 74.60659 74.06946 73.47527 72.70952 
##       41       42       43       44       45       46       47       48 
## 70.84526 72.29059 69.36420 71.32731 68.36805 68.22164 68.02480 67.80198 
##       49       50       51       52       53       54       55       56 
## 62.67439 61.78257 61.56393 60.57957 59.79209 58.63253 58.73899 58.26489 
##       57       58       59       60       61       62       63       64 
## 57.66446 57.00412 57.17968 56.70129 56.21848 55.59039 61.58557 61.07044 
##       65       66       67       68       69       70       71       72 
## 75.72930 74.91354 74.80593 74.68122 74.91998 74.87481 76.61278 74.45076 
##       73       74       75       76       77       78       79       80 
## 75.02599 75.01374 57.57648 55.71881 56.07674 57.29806 57.26066 57.19919 
##       81       82       83       84       85       86       87       88 
## 78.99514 78.82922 78.82007 80.62966 78.54490 77.68838 77.66873 77.46305 
##       89       90       91       92       93       94       95       96 
## 77.25752 77.01992 76.89438 79.04264 76.72813 76.76732 73.78286 73.58054 
##       97       98       99      100      101      102      103      104 
## 73.71721 75.68134 73.64809 73.62446 73.40452 72.95570 70.92440 72.28333 
##      105      106      107      108      109      110      111      112 
## 69.97095 69.25228 70.37677 71.28447 71.06114 70.97759 71.08924 71.11517 
##      113      114      115      116      117      118      119      120 
## 86.48200 87.57512 86.74027 86.53648 85.95242 85.17850 84.37249 84.62017 
##      121      122      123      124      125      126      127      128 
## 82.34892 84.82019 83.17514 81.46071 82.24857 82.62845 83.77267 80.45117 
##      129      130      131      132      133      134      135      136 
## 81.33991 81.80753 79.12155 82.46191 78.55990 78.47308 78.37476 77.84930 
##      137      138      139      140      141      142      143      144 
## 78.85891 75.12090 76.73171 76.41356 74.71749 78.38163 74.60178 74.43005 
##      145      146      147      148      149      150      151      152 
## 73.67048 73.62130 71.58070 72.10910 71.92615 72.37954 69.85783 69.77200 
##      153      154      155      156      157      158      159      160 
## 71.88268 68.44323 68.64389 68.49397 68.19152 67.93367 67.64084 69.78579 
##      161      162      163      164      165      166      167      168 
## 74.54986 76.46727 73.78245 74.15857 73.73845 73.80169 71.75403 71.42624 
##      169      170      171      172      173      174      175      176 
## 73.42342 73.35333 72.94236 73.03688 72.61853 75.49222 72.39311 66.32535 
##      177      178      179      180      181      182      183      184 
## 77.62952 78.85558 77.59169 77.50967 74.27751 74.33125 77.08383 76.17925 
##      185      186      187      188      189      190      191      192 
## 76.16830 75.87443 76.36237 76.11698 75.81362 74.97997 76.74089 75.13444 
##      193      194      195      196      197      198      199      200 
## 67.71985 67.42370 67.33406 67.13401 68.70534 66.03128 65.43963 65.44173 
##      201      202      203      204      205      206      207      208 
## 65.27457 64.95121 64.27718 64.18602 63.09961 62.79521 62.12925 61.76719 
##      209      210      211      212      213      214      215      216 
## 77.16286 78.88518 78.68463 76.22835 78.59669 74.10179 78.46929 73.54563 
##      217      218      219      220      221      222      223      224 
## 76.24802 73.58166 75.60563 75.25453 72.00829 72.05561 73.45453 72.68981 
##      225      226      227      228      229      230      231      232 
## 75.61369 75.68813 78.83244 73.26668 74.67746 74.50511 74.57422 74.24527 
##      233      234      235      236      237      238      239      240 
## 72.42351 73.12768 72.70687 72.40913 69.69105 71.54052 71.37708 71.25145 
##      241      242      243      244      245      246      247      248 
## 79.94802 81.53232 79.43854 79.30107 80.44355 78.90744 78.74574 81.22244 
##      249      250      251      252      253      254      255      256 
## 77.11156 78.64330 78.44050 82.86539 81.07864 81.02758 83.24702 81.86229 
##      257      258      259      260      261      262      263      264 
## 70.76349 70.75168 75.27054 71.87299 71.58038 71.73526 71.92659 71.94707 
##      265      266      267      268      269      270      271      272 
## 71.81601 71.25635 71.41873 71.01674 73.50723 69.77043 73.60502 69.02118 
##      273      274      275      276      277      278      279      280 
## 63.68413 63.47288 63.03403 62.67516 62.42736 62.05936 61.62574 60.93425 
##      281      282      283      284      285      286      287      288 
## 60.54367 60.34640 59.37800 58.99834 59.25930 58.83215 58.13934 58.00466 
##      289      290      291      292      293      294      295      296 
## 68.67593 68.43212 68.21340 67.90183 67.54452 62.12694 62.28517 61.98096 
##      297      298      299      300      301      302      303      304 
## 61.56178 60.95791 60.43300 57.48197 59.40416 56.73043 60.73882 57.85661 
##      305      306      307      308      309      310      311      312 
## 72.94357 72.82020 72.66072 70.92655 72.32770 73.99254 75.26251 69.70270 
##      313      314      315      316      317      318      319      320 
## 69.70185 69.64908 69.71349 69.74404 73.19371 69.08838 68.74774 68.16430 
##      321      322      323      324      325      326      327      328 
## 73.43800 73.44896 74.62784 73.97545 72.28066 72.40089 72.32233 74.17074 
##      329      330      331      332      333      334      335      336 
## 75.37966 74.49976 68.33391 65.80498 67.19871 66.96539 67.22116 55.72674 
##      337      338      339      340      341      342      343      344 
## 69.23326 68.94684 67.98234 72.14347 65.96313 64.95550 62.63612 59.99549 
##      345      346      347      348      349      350      351      352 
## 59.29333 57.47432 53.65952 48.12322 45.47941 43.30281 41.40884 41.95241 
##      353      354      355      356      357      358      359      360 
## 75.86418 75.40007 74.99455 74.89262 74.65765 74.24996 74.03441 71.87423 
##      361      362      363      364      365      366      367      368 
## 73.51120 73.34708 73.23640 76.09165 73.75479 73.57307 73.21450 72.93085 
##      369      370      371      372      373      374      375      376 
## 76.33977 76.32673 78.07563 77.80685 75.42001 76.94568 76.28913 76.83513 
##      377      378      379      380      381      382      383      384 
## 78.10061 74.84214 74.76280 74.37882 74.92517 74.56285 75.93672 75.16294 
##      385      386      387      388      389      390      391      392 
## 76.91438 74.58773 78.72751 76.32085 76.10801 75.32010 75.46989 75.38508 
##      393      394      395      396      397      398      399      400 
## 73.00809 72.81576 77.33277 74.56244 74.02528 73.91922 76.39690 73.61733 
##      401      402      403      404      405      406      407      408 
## 66.92343 62.51907 64.26087 59.37394 59.80268 59.31507 59.24422 59.43654 
##      409      410      411      412      413      414      415      416 
## 57.81792 62.70252 54.76583 52.71161 52.15238 51.05702 50.59361 50.22889 
##      417      418      419      420      421      422      423      424 
## 64.64857 64.46045 68.91370 63.61945 62.92205 67.20213 60.66628 63.75051 
##      425      426      427      428      429      430      431      432 
## 58.24793 55.85695 54.19474 53.26662 52.42892 52.04453 52.10241 51.76608 
##      433      434      435      436      437      438      439      440 
## 59.63626 66.14046 58.01697 59.53610 57.42722 58.18263 57.49144 57.27428 
##      441      442      443      444      445      446      447      448 
## 55.37505 55.29951 53.09141 54.77582 51.43814 50.28090 53.69246 53.10329 
##      449      450      451      452      453      454      455      456 
## 72.55781 71.48957 74.36196 71.82798 69.31326 71.36241 71.00054 70.75758 
##      457      458      459      460      461      462      463      464 
## 70.23870 70.00799 69.12060 71.20595 68.90973 68.81069 67.50052 61.99329 
##      465      466      467      468      469      470      471      472 
## 66.35086 66.12321 65.25677 65.02424 67.91971 64.53875 66.63710 66.39080 
##      473      474      475      476      477      478      479      480 
## 66.73954 62.98696 62.39027 61.56909 60.80547 59.55654 58.85203 58.30906 
##      481      482      483      484      485      486      487      488 
## 60.92966 60.40917 60.23432 59.79064 59.17756 64.59995 57.71742 56.99515 
##      489      490      491      492      493      494      495      496 
## 55.81637 55.85586 62.79646 55.32852 55.13636 61.77281 60.23915 53.31214 
##      497      498      499      500      501      502      503      504 
## 81.65569 78.77230 81.19940 81.15948 76.48336 80.03223 77.72248 79.07597 
##      505      506      507      508      509      510      511      512 
## 79.37638 78.81541 80.25553 78.00873 76.32444 77.09182 78.25334 76.96047 
##      513      514      515      516      517      518      519      520 
## 55.52328 54.50360 54.09848 54.25904 52.89461 52.21921 52.16508 51.13457 
##      521      522      523      524      525      526      527      528 
## 50.31898 57.40109 48.59126 47.95208 56.35227 55.73487 55.40393 55.19733 
##      529      530      531      532      533      534      535      536 
## 57.14992 56.89553 56.73111 56.38264 55.54103 55.50279 60.69656 53.91465 
##      537      538      539      540      541      542      543      544 
## 53.93446 53.27362 59.88893 59.76166 59.41331 58.99225 59.18109 56.51258 
##      545      546      547      548      549      550      551      552 
## 79.15416 79.03483 77.05941 75.96394 77.44647 75.20280 77.21081 77.06617 
##      553      554      555      556      557      558      559      560 
## 78.04703 77.00799 76.60329 75.88429 77.50340 77.10903 76.79312 76.41055 
##      561      562      563      564      565      566      567      568 
## 73.18990 72.87582 71.57202 71.85462 72.93894 71.77703 71.35285 69.36613 
##      569      570      571      572      573      574      575      576 
## 69.22263 68.81272 65.61534 67.68592 67.31618 67.23831 66.51250 64.19678 
##      577      578      579      580      581      582      583      584 
## 74.01847 71.98883 73.59797 73.87138 71.21955 72.98456 75.19558 72.60063 
##      585      586      587      588      589      590      591      592 
## 72.25647 71.60556 69.71976 68.89808 73.94354 68.82017 68.67765 68.50603 
##      593      594      595      596      597      598      599      600 
## 66.99429 68.26519 64.28204 64.08726 64.03460 63.85008 63.39374 62.60985 
##      601      602      603      604      605      606      607      608 
## 62.29637 62.74131 62.33460 58.86371 58.55158 59.62815 57.89333 57.67509 
##      609      610      611      612      613      614      615      616 
## 63.57089 63.31488 63.28335 62.15987 62.15925 61.53156 60.93347 59.98824 
##      617      618      619      620      621      622      623      624 
## 59.07178 58.31506 57.38389 63.14025 55.76006 55.46198 56.17195 56.16314 
##      625      626      627      628      629      630      631      632 
## 72.16426 74.02143 75.70432 75.47074 74.27878 73.07850 75.78243 72.34805 
##      633      634      635      636      637      638      639      640 
## 72.31713 71.92444 73.23788 72.24639 72.81169 72.87703 73.07127 74.70941 
##      641      642      643      644      645      646      647      648 
## 70.28922 78.52182 78.33822 78.14377 79.33295 79.23558 76.32878 76.85722 
##      649      650      651      652      653      654      655      656 
## 77.18400 77.00964 76.48698 75.64176 75.74096 74.79596 74.44044 74.34325 
##      657      658      659      660      661      662      663      664 
## 74.15706 75.73560 73.86782 76.22492 77.25141 77.29949 78.27883 80.40104 
##      665      666      667      668      669      670      671      672 
## 79.28403 78.20090 76.37052 75.80892 74.59345 74.36387 74.90325 71.29483 
##      673      674      675      676      677      678      679      680 
## 72.70337 76.17672 77.86891 77.68022 78.88049 78.98843 78.22704 78.71140 
##      681      682      683      684      685      686      687      688 
## 77.33324 78.39685 78.01634 77.95242 77.76916 76.34219 76.46036 75.58025 
##      689      690      691      692      693      694      695      696 
## 77.35914 79.70133 79.25400 81.72096 78.82313 77.85041 78.97222 80.45046 
##      697      698      699      700      701      702      703      704 
## 79.78945 79.08310 79.32343 77.37756 73.57404 73.08492 79.46348 76.13492 
##      705      706      707      708      709      710      711      712 
## 75.59429 71.68432 71.14399 70.79531 69.28203 68.53047 68.40481 65.05712 
##      713      714      715      716      717      718      719      720 
## 63.65669 68.93664 69.35268 69.51000 62.12093 66.54114 66.78592 66.72872 
##      721      722      723      724      725      726      727      728 
## 64.59937 60.37143 60.09171 56.61620 53.55242 57.80683 58.42780 58.97923 
##      729      730      731      732      733      734      735      736 
## 56.89766 53.07332 55.83168 51.28344 53.96944 50.81960 52.98849 53.80545 
##      737      738      739      740      741      742      743      744 
## 48.09027 82.29626 84.67202 84.64690 81.23050 82.57495 77.84797 80.29177 
##      745      746      747      748      749      750      751      752 
## 80.00961 79.98029 79.88949 81.70391 81.35187 80.61853 82.09712 80.27032 
##      753      754      755      756      757      758      759      760 
## 80.44783 60.31090 59.74384 60.08795 59.98082 59.04113 58.41722 58.01413 
##      761      762      763      764      765      766      767      768 
## 57.47371 55.68563 59.94693 54.49290 54.98520 54.62419 54.36484 53.93905 
##      769      770      771      772      773      774      775      776 
## 53.97994 74.65149 71.22406 71.97109 70.65629 70.43158 72.95010 70.02235 
##      777      778      779      780      781      782      783      784 
## 69.94634 69.41405 67.66307 68.72945 71.18765 67.62853 70.89027 67.60814 
##      785      786      787      788      789      790      791      792 
## 67.62757 67.57878 72.46842 72.79928 71.83345 71.51472 71.47803 70.52343 
##      793      794      795      796      797      798      799      800 
## 71.83207 73.30397 75.18413 72.68769 71.06914 70.06793 71.32971 69.65626 
##      801      802      803      804      805      806      807      808 
## 69.35162 69.25773 73.43724 71.11666 72.91326 72.41702 72.17087 71.82017 
##      809      810      811      812      813      814      815      816 
## 71.55144 71.42077 71.18442 70.84083 70.78732 70.48569 70.50993 70.37946 
##      817      818      819      820      821      822      823      824 
## 70.19542 68.48971 72.95688 72.74386 72.66229 72.59233 69.74351 69.71284 
##      825      826      827      828      829      830      831      832 
## 74.93917 72.13106 73.11773 69.76504 68.92327 68.51126 73.77232 71.13540 
##      833      834      835      836      837      838      839      840 
## 69.88365 69.56725 65.36530 65.22466 59.43628 58.93292 58.97801 59.17071 
##      841      842      843      844      845      846      847      848 
## 63.79017 59.69652 58.74183 58.75104 58.01712 59.37369 59.82813 59.86146 
##      849      850      851      852      853      854      855      856 
## 59.76958 49.66155 60.80671 60.62411 60.44475 60.19947 59.91761 57.28423 
##      857      858      859      860      861      862      863      864 
## 59.36975 59.16914 58.97542 58.77914 61.07826 55.16341 54.85344 53.70620 
##      865      866      867      868      869      870      871      872 
## 52.73770 48.03555 77.93002 78.67742 77.68457 78.21624 77.95061 77.64625 
##      873      874      875      876      877      878      879      880 
## 77.26561 76.41470 76.59400 76.34270 75.54405 75.18454 75.26254 74.49881 
##      881      882      883      884      885      886      887      888 
## 74.03330 73.35058 61.94753 61.81040 61.66348 61.45594 61.00468 59.82729 
##      889      890      891      892      893      894      895      896 
## 59.30664 58.26763 62.05325 56.11884 54.85810 54.03859 53.54657 52.97607 
##      897      898      899      900      901      902      903      904 
## 52.16670 49.35647 74.79256 77.41177 74.27343 72.09226 73.81387 77.10628 
##      905      906      907      908      909      910      911      912 
## 76.60115 76.34429 75.99042 72.70176 72.55697 71.97742 75.29829 75.22604 
##      913      914      915      916      917      918      919      920 
## 71.42589 68.77760 79.73187 81.84177 81.77063 81.56063 77.12934 77.16805 
##      921      922      923      924      925      926      927      928 
## 79.25737 79.23890 81.33435 79.14297 82.19915 83.06668 81.10212 80.67843 
##      929      930      931      932      933      934      935      936 
## 81.98939 81.29448 80.80087 80.98556 80.89463 78.93627 76.62264 76.44253 
##      937      938      939      940      941      942      943      944 
## 80.39777 80.14117 77.92022 79.56048 78.90278 78.70933 78.36049 79.60228 
##      945      946      947      948      949      950      951      952 
## 80.06964 79.89122 67.03684 66.82986 65.92737 64.84300 63.63265 63.35676 
##      953      954      955      956      957      958      959      960 
## 67.00992 61.70931 61.64325 60.96867 64.57417 59.26206 65.39426 60.21358 
##      961      962      963      964      965      966      967      968 
## 65.48546 61.15917 63.94205 63.85791 63.78649 63.65188 63.21784 62.94930 
##      969      970      971      972      973      974      975      976 
## 62.61247 62.15132 61.82559 61.56814 60.46429 57.85799 58.21351 58.00380 
##      977      978      979      980      981      982      983      984 
## 63.76225 63.01504 74.92351 74.55832 74.33770 76.45150 72.85035 72.25983 
##      985      986      987      988      989      990      991      992 
## 72.16542 70.85694 73.88879 70.91176 70.36019 69.90250 69.83892 69.59525 
##      993      994      995      996      997      998      999     1000 
## 69.79029 64.30840 82.52566 80.81623 80.63755 78.18132 77.98142 82.05265 
##     1001     1002     1003     1004     1005     1006     1007     1008 
## 81.98625 82.79978 81.23479 80.77554 80.55899 80.44983 80.21010 78.56155 
##     1009     1010     1011     1012     1013     1014     1015     1016 
## 79.37171 78.46042 65.23752 67.61889 66.46665 66.98632 66.37495 65.61746 
##     1017     1018     1019     1020     1021     1022     1023     1024 
## 65.24602 64.56017 68.06908 60.53009 59.67497 63.85432 63.71162 62.64738 
##     1025     1026     1027     1028     1029     1030     1031     1032 
## 62.77167 62.87689 79.72265 80.64119 80.54149 80.25398 80.37161 80.12251 
##     1033     1034     1035     1036     1037     1038     1039     1040 
## 78.77669 79.89679 81.34165 78.70635 76.97659 78.53465 78.22753 76.83181 
##     1041     1042     1043     1044     1045     1046     1047     1048 
## 75.46880 73.63889 75.95160 74.36145 75.69241 75.60206 77.55261 69.82812 
##     1049     1050     1051     1052     1053     1054     1055     1056 
## 69.56351 69.31153 69.14472 68.88780 67.33860 65.81639 68.26693 68.04476 
##     1057     1058     1059     1060     1061     1062     1063     1064 
## 67.72430 67.63726 66.06522 67.55500 67.46900 69.74226 67.16120 69.36961 
##     1065     1066     1067     1068     1069     1070     1071     1072 
## 69.18384 72.30007 66.52460 69.22296 68.80495 68.72197 68.37773 68.15277 
##     1073     1074     1075     1076     1077     1078     1079     1080 
## 64.52684 62.69363 61.16652 60.73317 60.57459 59.96626 64.32613 58.75077 
##     1081     1082     1083     1084     1085     1086     1087     1088 
## 58.33618 63.43908 63.02738 57.42350 56.66372 55.16342 55.69245 55.28853 
##     1089     1090     1091     1092     1093     1094     1095     1096 
## 55.44761 55.18823 60.57284 60.28946 59.67025 59.13755 58.72593 58.56990 
##     1097     1098     1099     1100     1101     1102     1103     1104 
## 58.43709 58.12541 62.90094 61.73513 58.62698 59.10417 59.06338 59.13063 
##     1105     1106     1107     1108     1109     1110     1111     1112 
## 59.68287 60.16327 68.84849 68.58889 68.69536 72.42967 68.37492 68.20536 
##     1113     1114     1115     1116     1117     1118     1119     1120 
## 68.66503 67.24759 66.78356 72.45555 68.00174 67.73309 66.65971 67.17819 
##     1121     1122     1123     1124     1125     1126     1127     1128 
## 64.97995 65.23339 67.95345 63.93344 63.74875 63.37699 62.31017 54.87582 
##     1129     1130     1131     1132     1133     1134     1135     1136 
## 62.23528 61.70240 60.03309 65.12749 64.62634 64.30193 64.59799 63.83727 
##     1137     1138     1139     1140     1141     1142     1143     1144 
## 63.58959 63.43364 71.11725 69.57612 73.65667 71.23477 71.06928 70.86168 
##     1145     1146     1147     1148     1149     1150     1151     1152 
## 70.59822 70.25788 72.47883 69.66597 69.35116 68.93295 68.69181 66.99743 
##     1153     1154     1155     1156     1157     1158     1159     1160 
## 67.93730 66.23613 78.14249 78.31554 77.85199 77.67137 79.42787 76.76884 
##     1161     1162     1163     1164     1165     1166     1167     1168 
## 77.17582 75.21828 76.86321 76.59880 76.28017 78.94596 75.90763 75.15964 
##     1169     1170     1171     1172     1173     1174     1175     1176 
## 74.73939 74.49855 82.64869 80.52578 81.28665 82.50083 81.76144 83.56310 
##     1177     1178     1179     1180     1181     1182     1183     1184 
## 81.74033 83.93698 84.48130 81.33029 83.54305 83.30861 80.21575 82.63242 
##     1185     1186     1187     1188     1189     1190     1191     1192 
## 81.13453 80.95253 63.58529 63.69838 64.71021 67.35584 63.48156 63.03962 
##     1193     1194     1195     1196     1197     1198     1199     1200 
## 65.98761 65.52893 65.15116 64.46466 61.35683 60.73830 60.26976 60.12835 
##     1201     1202     1203     1204     1205     1206     1207     1208 
## 59.72036 59.84982 68.83134 68.68852 68.51490 68.17611 67.67981 67.44324 
##     1209     1210     1211     1212     1213     1214     1215     1216 
## 66.90201 66.89498 68.67882 66.12281 69.29987 65.47391 65.68693 65.02483 
##     1217     1218     1219     1220     1221     1222     1223     1224 
## 65.17906 65.14743 76.26077 76.17475 76.24410 74.97108 74.21876 75.48305 
##     1225     1226     1227     1228     1229     1230     1231     1232 
## 73.32815 72.30997 70.24289 71.43405 71.38182 71.63764 70.99104 71.30282 
##     1233     1234     1235     1236     1237     1238     1239     1240 
## 70.89961 73.07664 67.54897 67.16056 70.87479 68.63582 68.39470 68.08656 
##     1241     1242     1243     1244     1245     1246     1247     1248 
## 67.54113 67.60847 66.44039 65.91161 69.73197 69.33079 65.72352 67.43105 
##     1249     1250     1251     1252     1253     1254     1255     1256 
## 64.62366 66.19133 81.96396 81.97707 81.87124 81.83394 85.12598 83.59462 
##     1257     1258     1259     1260     1261     1262     1263     1264 
## 83.39871 83.79787 83.44054 82.95576 78.13161 79.65631 77.23229 78.14166 
##     1265     1266     1267     1268     1269     1270     1271     1272 
## 77.76257 77.31778 80.79582 81.77335 80.61651 81.37236 80.29165 78.72107 
##     1273     1274     1275     1276     1277     1278     1279     1280 
## 79.74122 79.92408 78.33515 78.40546 76.44199 77.33784 79.28537 77.67373 
##     1281     1282     1283     1284     1285     1286     1287     1288 
## 77.50869 77.56644 80.01725 81.61003 80.20003 82.69154 82.55313 82.49181 
##     1289     1290     1291     1292     1293     1294     1295     1296 
## 79.49795 79.62173 81.17133 79.35702 80.49657 80.22270 79.50564 79.07363 
##     1297     1298     1299     1300     1301     1302     1303     1304 
## 77.80622 76.68760 73.75677 73.40800 71.64459 73.05686 72.91584 71.43878 
##     1305     1306     1307     1308     1309     1310     1311     1312 
## 70.54802 72.63625 72.17624 74.33920 71.25234 69.96960 70.55946 70.33312 
##     1313     1314     1315     1316     1317     1318     1319     1320 
## 69.58624 69.83824 79.29186 77.70127 77.69939 77.63447 79.42046 77.70917 
##     1321     1322     1323     1324     1325     1326     1327     1328 
## 77.60894 78.96195 78.89808 78.75557 78.79203 78.54347 77.94555 77.28889 
##     1329     1330     1331     1332     1333     1334     1335     1336 
## 76.28947 76.87232 74.48663 74.42493 74.53399 74.18415 74.33639 74.54605 
##     1337     1338     1339     1340     1341     1342     1343     1344 
## 72.92064 72.80693 76.57304 74.66713 74.47973 74.23253 73.95023 73.71595 
##     1345     1346     1347     1348     1349     1350     1351     1352 
## 73.37404 73.17027 74.36561 77.45257 77.26712 72.52097 73.64254 72.89861 
##     1353     1354     1355     1356     1357     1358     1359     1360 
## 73.25636 72.85709 72.24197 71.68793 70.97427 69.93599 70.42875 70.15408 
##     1361     1362     1363     1364     1365     1366     1367     1368 
## 69.48997 69.18250 63.41793 65.63791 63.89101 64.64626 64.70077 61.53351 
##     1369     1370     1371     1372     1373     1374     1375     1376 
## 60.33504 58.97394 57.05741 55.45065 54.25660 52.45537 51.09462 50.72843 
##     1377     1378     1379     1380     1381     1382     1383     1384 
## 49.86350 49.19628 69.11118 72.52428 74.43792 74.33754 74.31278 74.26489 
##     1385     1386     1387     1388     1389     1390     1391     1392 
## 72.36371 72.20293 71.38004 68.62985 64.34520 64.39516 61.87318 65.59364 
##     1393     1394     1395     1396     1397     1398     1399     1400 
## 63.06497 62.62252 77.03090 75.06707 75.38901 77.83874 77.81051 77.37303 
##     1401     1402     1403     1404     1405     1406     1407     1408 
## 77.41342 78.29445 77.88872 77.88963 77.52094 77.35798 75.89168 76.43810 
##     1409     1410     1411     1412     1413     1414     1415     1416 
## 75.71628 76.50497 71.76450 74.52541 71.51023 73.97614 70.63689 67.77306 
##     1417     1418     1419     1420     1421     1422     1423     1424 
## 72.54321 70.35712 70.12939 69.95399 69.97542 70.05386 69.75013 69.73467 
##     1425     1426     1427     1428     1429     1430     1431     1432 
## 69.34548 68.79760 65.45782 64.42776 67.27610 67.56655 63.90039 63.54985 
##     1433     1434     1435     1436     1437     1438     1439     1440 
## 63.28876 62.86902 62.54967 66.25129 62.61233 62.32447 61.94666 61.46152 
##     1441     1442     1443     1444     1445     1446     1447     1448 
## 61.32826 61.09846 77.93376 75.91568 74.58655 77.16562 76.67690 80.04494 
##     1449     1450     1451     1452     1453     1454     1455     1456 
## 77.48145 77.29346 76.07291 76.24617 75.97788 78.87274 78.37677 74.50587 
##     1457     1458     1459     1460     1461     1462     1463     1464 
## 73.68909 73.35510 72.69232 72.95062 72.62074 73.08416 72.96508 72.96762 
##     1465     1466     1467     1468     1469     1470     1471     1472 
## 72.40344 72.30467 72.05768 74.23223 66.96505 67.33686 69.72452 69.58809 
##     1473     1474     1475     1476     1477     1478     1479     1480 
## 69.52516 67.62323 56.80440 55.99679 56.62329 57.27237 64.67891 55.07489 
##     1481     1482     1483     1484     1485     1486     1487     1488 
## 50.87852 44.91128 42.68486 40.28298 39.47361 39.72797 40.84627 41.84215 
##     1489     1490     1491     1492     1493     1494     1495     1496 
## 42.35921 43.17951 62.44135 61.11215 62.52716 61.79521 61.88462 61.65138 
##     1497     1498     1499     1500     1501     1502     1503     1504 
## 61.59899 61.26568 60.39141 64.86429 60.10247 59.61041 64.78037 58.49497 
##     1505     1506     1507     1508     1509     1510     1511     1512 
## 59.32712 62.66196 73.84287 73.76372 73.96276 73.99948 72.05776 74.69533 
##     1513     1514     1515     1516     1517     1518     1519     1520 
## 74.89322 75.62367 75.19710 75.73022 75.78745 75.48594 75.73340 75.20780 
##     1521     1522     1523     1524     1525     1526     1527     1528 
## 75.40563 75.15714 78.25034 78.27189 77.91661 75.95804 78.89668 75.84517 
##     1529     1530     1531     1532     1533     1534     1535     1536 
## 77.55098 76.94564 80.21276 75.95713 76.16068 77.92088 78.92227 78.60455 
##     1537     1538     1539     1540     1541     1542     1543     1544 
## 77.85606 77.68914 77.94708 81.52596 81.21318 77.63445 83.19642 78.11262 
##     1545     1546     1547     1548     1549     1550     1551     1552 
## 77.85501 83.85684 77.00976 81.24218 78.62794 80.32349 79.84491 76.79794 
##     1553     1554     1555     1556     1557     1558     1559     1560 
## 78.64828 78.65526 67.05455 63.35798 67.57063 63.63743 63.53107 63.35542 
##     1561     1562     1563     1564     1565     1566     1567     1568 
## 62.80403 62.35893 61.99905 61.82847 61.66070 60.70118 59.88583 60.67960 
##     1569     1570     1571     1572     1573     1574     1575     1576 
## 60.46721 56.34584 59.85591 60.33146 57.98286 66.13201 57.41080 51.19882 
##     1577     1578     1579     1580     1581     1582     1583     1584 
## 53.37170 51.47793 47.71008 48.50833 57.12669 46.09834 44.43615 54.09995 
##     1585     1586     1587     1588     1589     1590     1591     1592 
## 44.20510 44.74047 72.43191 73.61630 72.97732 72.84831 74.84831 72.53615 
##     1593     1594     1595     1596     1597     1598     1599     1600 
## 72.58205 72.39563 72.03806 72.21296 71.27356 71.90450 71.40346 73.44200 
##     1601     1602     1603     1604     1605     1606     1607     1608 
## 70.96891 70.28446 73.16191 73.26866 72.71415 72.22707 71.76656 71.09378 
##     1609     1610     1611     1612     1613     1614     1615     1616 
## 71.22495 70.60420 70.80102 70.91586 70.93549 72.14957 69.95595 69.66552 
##     1617     1618     1619     1620     1621     1622     1623     1624 
## 69.34914 68.88376 60.90746 60.56944 60.23663 64.35021 59.77151 58.74074 
##     1625     1626     1627     1628     1629     1630     1631     1632 
## 58.40014 57.82368 58.17058 57.61639 61.81532 57.33710 56.80261 61.17977 
##     1633     1634     1635     1636     1637     1638     1639     1640 
## 60.64342 59.72677 79.82402 78.51259 78.23831 78.06869 79.59475 75.91460 
##     1641     1642     1643     1644     1645     1646     1647     1648 
## 75.71673 76.74452 76.40252 76.58748 78.81814 75.18209 77.69538 77.44917 
##     1649     1650     1651     1652     1653     1654     1655     1656 
## 76.70200 77.97695 62.77687 65.20438 65.24762 65.70507 65.25960 61.82261 
##     1657     1658     1659     1660     1661     1662     1663     1664 
## 61.58665 61.12641 61.09293 61.14340 60.86650 60.79522 60.81782 60.66750 
##     1665     1666     1667     1668     1669     1670     1671     1672 
## 62.05733 60.51792 64.18962 74.72026 73.80828 74.18566 73.63481 73.34679 
##     1673     1674     1675     1676     1677     1678     1679     1680 
## 71.62409 71.72586 72.11434 72.32916 72.06703 71.58028 71.33402 70.92351 
##     1681     1682     1683     1684     1685     1686     1687     1688 
## 68.34961 70.36475 67.86922 72.74342 72.12545 73.88864 74.02015 72.34699 
##     1689     1690     1691     1692     1693     1694     1695     1696 
## 72.10232 76.01015 73.86692 73.42389 73.58833 73.50376 73.34428 73.03638 
##     1697     1698     1699     1700     1701     1702     1703     1704 
## 72.41064 72.36518 72.05133 69.98172 70.12653 70.00448 69.92680 69.91237 
##     1705     1706     1707     1708     1709     1710     1711     1712 
## 69.71122 70.46716 69.30521 69.03741 68.87430 71.03326 67.92361 71.93616 
##     1713     1714     1715     1716     1717     1718     1719     1720 
## 70.68383 67.75620 55.16414 75.70449 73.13006 73.37868 71.50394 72.96525 
##     1721     1722     1723     1724     1725     1726     1727     1728 
## 72.67275 75.92463 71.80136 71.56698 74.65067 69.86453 69.02275 68.22011 
##     1729     1730     1731     1732     1733     1734     1735     1736 
## 66.46984 67.46235 66.54547 66.09093 76.20449 78.49953 76.43989 76.55574 
##     1737     1738     1739     1740     1741     1742     1743     1744 
## 76.66401 76.45201 76.05635 75.55710 74.86936 74.08579 72.45915 73.48430 
##     1745     1746     1747     1748     1749     1750     1751     1752 
## 55.82672 57.95068 57.91563 57.75840 72.90513 72.82432 72.72324 72.04864 
##     1753     1754     1755     1756     1757     1758     1759     1760 
## 73.12265 72.79978 70.66718 70.27283 68.18943 69.65106 69.28892 68.89339 
##     1761     1762     1763     1764     1765     1766     1767     1768 
## 68.37905 70.28960 67.46147 69.33020 58.78253 58.35308 63.60864 62.73558 
##     1769     1770     1771     1772     1773     1774     1775     1776 
## 61.02330 60.13325 60.55376 59.40411 51.35922 49.90394 49.15362 49.07899 
##     1777     1778     1779     1780     1781     1782     1783     1784 
## 48.74940 49.22939 55.84288 56.14338 63.53608 66.66875 66.49400 66.28519 
##     1785     1786     1787     1788     1789     1790     1791     1792 
## 65.49868 65.79543 62.17636 60.27047 61.45420 64.77791 61.06715 62.22768 
##     1793     1794     1795     1796     1797     1798     1799     1800 
## 60.58140 60.40864 60.18939 59.98295 67.95256 65.77105 65.58885 64.83933 
##     1801     1802     1803     1804     1805     1806     1807     1808 
## 63.36196 62.67327 66.01292 57.55086 55.99926 52.86753 50.69995 49.58756 
##     1809     1810     1811     1812     1813     1814     1815     1816 
## 49.32552 49.67039 50.35325 57.47097 70.33218 67.29795 69.01703 68.80275 
##     1817     1818     1819     1820     1821     1822     1823     1824 
## 66.21328 68.19745 64.92236 65.63664 63.67113 63.40419 67.46085 65.62196 
##     1825     1826     1827     1828     1829     1830     1831     1832 
## 61.95096 65.29314 60.99602 65.01904 61.05443 83.70997 83.68764 85.00037 
##     1833     1834     1835     1836     1837     1838     1839     1840 
## 81.94633 80.47733 80.24071 79.87791 82.84582 82.34032 81.80889 81.52405 
##     1841     1842     1843     1844     1845     1846     1847     1848 
## 81.32814 82.26270 80.93930 80.57007 80.24093 82.77704 82.72161 84.58149 
##     1849     1850     1851     1852     1853     1854     1855     1856 
## 83.94614 84.52782 84.76437 82.29035 81.04128 79.74146 79.60593 79.36479 
##     1857     1858     1859     1860     1861     1862     1863     1864 
## 78.88789 77.58946 76.70912 78.37101 78.07018 71.86798 71.77059 71.40159 
##     1865     1866     1867     1868     1869     1870     1871     1872 
## 71.33458 69.70102 69.42269 70.74493 70.41802 70.17886 70.41786 67.23972 
##     1873     1874     1875     1876     1877     1878     1879     1880 
## 67.12359 66.98915 66.61994 66.36121 66.06380 62.25878 58.66182 59.14555 
##     1881     1882     1883     1884     1885     1886     1887     1888 
## 58.76839 58.30225 57.65441 56.43583 55.90588 55.58926 55.38958 55.01107 
##     1889     1890     1891     1892     1893     1894     1895     1896 
## 53.42523 57.90795 53.68127 53.05907 53.67798 60.05219 60.31809 59.10012 
##     1897     1898     1899     1900     1901     1902     1903     1904 
## 59.40369 58.19823 58.16072 57.90603 57.39400 56.60705 56.81607 61.55346 
##     1905     1906     1907     1908     1909     1910     1911     1912 
## 61.83455 56.19488 57.50883 55.47835 54.43451 75.55101 82.00653 79.45623 
##     1913     1914     1915     1916     1917     1918     1919     1920 
## 79.23893 81.76145 81.14395 85.10189 80.91431 81.42248 85.01924 85.42600 
##     1921     1922     1923     1924     1925     1926     1927     1928 
## 83.96663 80.85756 80.44941 80.30771 82.18010 80.78887 75.54209 76.41022 
##     1929     1930     1931     1932     1933     1934     1935     1936 
## 77.23605 77.21864 76.19820 76.05683 75.75244 74.46674 70.71020 70.68491 
##     1937     1938     1939     1940     1941     1942     1943     1944 
## 72.66134 71.98177 71.56340 73.70976 71.23121 65.25482 63.15032 62.79318 
##     1945     1946     1947     1948     1949     1950     1951     1952 
## 62.48705 62.28178 63.46560 63.25874 63.00474 62.28478 61.68213 60.76199 
##     1953     1954     1955     1956     1957     1958     1959     1960 
## 63.51657 59.72414 59.57795 59.39208 59.16499 62.11241 78.20699 72.76947 
##     1961     1962     1963     1964     1965     1966     1967     1968 
## 72.70416 72.54950 73.70515 73.16806 74.33424 71.79010 71.66359 71.33360 
##     1969     1970     1971     1972     1973     1974     1975     1976 
## 73.03325 70.87939 73.77135 73.61317 73.39673 73.03655 72.70858 64.31285 
##     1977     1978     1979     1980     1981     1982     1983     1984 
## 64.17437 63.98624 63.68206 63.60210 63.21519 62.78015 66.98840 61.85830 
##     1985     1986     1987     1988     1989     1990     1991     1992 
## 61.41722 65.72762 59.28729 60.00025 59.37853 59.25520 58.71018 69.26677 
##     1993     1994     1995     1996     1997     1998     1999     2000 
## 70.33276 69.77801 69.76240 72.49038 69.96654 69.61317 69.14092 71.49164 
##     2001     2002     2003     2004     2005     2006     2007     2008 
## 71.35263 71.43874 71.11687 69.32220 67.52400 71.17047 68.26475 71.92409 
##     2009     2010     2011     2012     2013     2014     2015     2016 
## 72.11275 72.04941 74.24509 74.02791 75.71472 75.79712 71.59029 71.77371 
##     2017     2018     2019     2020     2021     2022     2023     2024 
## 73.11153 70.46929 70.15816 72.75528 73.35551 70.60083 72.75613 67.51778 
##     2025     2026     2027     2028     2029     2030     2031     2032 
## 66.34069 67.11953 67.14911 66.93936 66.78702 67.05874 69.25046 66.72942 
##     2033     2034     2035     2036     2037     2038     2039     2040 
## 66.06361 66.00912 66.55109 66.36883 66.11952 66.13252 66.09617 79.09451 
##     2041     2042     2043     2044     2045     2046     2047     2048 
## 80.96310 77.47155 77.86383 77.87099 79.54711 77.26241 78.73515 76.37475 
##     2049     2050     2051     2052     2053     2054     2055     2056 
## 76.25174 76.07023 76.91511 76.68138 76.30399 75.90198 75.64077 79.57556 
##     2057     2058     2059     2060     2061     2062     2063     2064 
## 77.70179 78.12227 79.09788 79.45283 79.12712 79.51425 79.32691 77.79999 
##     2065     2066     2067     2068     2069     2070     2071     2072 
## 78.54986 79.97927 77.96899 79.98886 79.82913 78.01112 77.28136 79.33503 
##     2073     2074     2075     2076     2077     2078     2079     2080 
## 79.31643 77.48874 80.08534 79.03590 74.61731 77.31947 78.14478 74.94046 
##     2081     2082     2083     2084     2085     2086     2087     2088 
## 76.12308 78.27757 77.65021 76.65078 74.52817 74.77719 75.40081 77.92914 
##     2089     2090     2091     2092     2093     2094     2095     2096 
## 77.27097 76.63965 76.40950 76.41089 77.10431 77.75247 77.69053 77.27661 
##     2097     2098     2099     2100     2101     2102     2103     2104 
## 75.91095 76.19937 71.73455 75.09738 74.06098 76.36143 72.47528 70.56301 
##     2105     2106     2107     2108     2109     2110     2111     2112 
## 71.41432 70.97114 73.28654 70.16663 62.81347 68.87409 70.77485 70.04549 
##     2113     2114     2115     2116     2117     2118     2119     2120 
## 70.42779 70.82814 70.65842 71.01787 70.03548 70.08595 68.23851 72.53474 
##     2121     2122     2123     2124     2125     2126     2127     2128 
## 76.21858 75.11952 76.83320 74.41626 77.03173 76.22283 75.51425 75.32608 
##     2129     2130     2131     2132     2133     2134     2135     2136 
## 74.70679 74.18542 73.66156 73.03781 72.41304 72.06525 71.89956 72.84244 
##     2137     2138     2139     2140     2141     2142     2143     2144 
## 74.85035 74.56699 73.99959 73.57001 72.44076 72.90578 72.54337 71.74775 
##     2145     2146     2147     2148     2149     2150     2151     2152 
## 76.61998 70.93875 70.62261 70.29376 70.17110 74.60857 74.28368 66.87084 
##     2153     2154     2155     2156     2157     2158     2159     2160 
## 69.81450 66.00649 66.10967 65.32445 68.52055 63.65107 63.38418 62.20298 
##     2161     2162     2163     2164     2165     2166     2167     2168 
## 59.86831 64.38642 54.67592 62.45794 53.01401 52.40322 51.87391 73.51837 
##     2169     2170     2171     2172     2173     2174     2175     2176 
## 73.23830 73.31946 73.26266 73.27256 73.00394 72.87343 72.62966 72.37554 
##     2177     2178     2179     2180     2181     2182     2183     2184 
## 72.46286 68.33897 74.08590 71.47275 70.57510 69.07490 71.36953 63.85532 
##     2185     2186     2187     2188     2189     2190     2191     2192 
## 73.47245 73.34016 73.09659 73.31449 74.01654 72.88956 72.76642 72.70931 
##     2193     2194     2195     2196     2197     2198     2199     2200 
## 75.20976 72.32815 72.28243 71.87875 71.61543 71.55345 71.05121 64.80471 
##     2201     2202     2203     2204     2205     2206     2207     2208 
## 72.27503 72.16175 72.06858 71.86242 71.77039 71.61551 70.94082 71.31570 
##     2209     2210     2211     2212     2213     2214     2215     2216 
## 68.94425 68.62578 73.31174 70.46741 70.10394 69.87294 69.68358 74.77597 
##     2217     2218     2219     2220     2221     2222     2223     2224 
## 75.13546 71.04075 67.85665 68.74658 68.19070 67.71066 70.78288 70.17293 
##     2225     2226     2227     2228     2229     2230     2231     2232 
## 66.88609 66.63893 69.96924 65.60561 65.80970 65.69877 65.12783 63.42327 
##     2233     2234     2235     2236     2237     2238     2239     2240 
## 62.13132 77.81134 78.59784 77.97745 77.66352 75.55963 75.74819 74.62742 
##     2241     2242     2243     2244     2245     2246     2247     2248 
## 74.19845 74.75470 74.33131 73.92927 75.57017 75.35809 75.17009 75.01769 
##     2249     2250     2251     2252     2253     2254     2255     2256 
## 74.91810 63.90478 63.47489 63.89358 67.51133 66.67809 62.21666 61.81621 
##     2257     2258     2259     2260     2261     2262     2263     2264 
## 60.62217 65.52827 59.87020 59.55348 59.24086 58.80622 58.26270 57.79389 
##     2265     2266     2267     2268     2269     2270     2271     2272 
## 62.00025 73.74016 75.04072 75.13580 75.39102 74.89814 74.47284 74.81584 
##     2273     2274     2275     2276     2277     2278     2279     2280 
## 73.97970 74.11274 74.29965 73.97416 72.86170 71.26005 73.76464 73.45245 
##     2281     2282     2283     2284     2285     2286     2287     2288 
## 73.10365 73.47361 75.10493 73.21160 73.02474 72.60559 74.73290 71.75175 
##     2289     2290     2291     2292     2293     2294     2295     2296 
## 72.17409 72.09471 70.55433 69.62966 70.75289 70.95046 70.87186 70.90246 
##     2297     2298     2299     2300     2301     2302     2303     2304 
## 65.60508 59.82485 58.89737 68.27629 61.45158 58.54257 57.48755 57.66869 
##     2305     2306     2307     2308     2309     2310     2311     2312 
## 55.89830 62.63269 55.59004 63.30591 55.08589 62.73573 54.41007 54.81570 
##     2313     2314     2315     2316     2317     2318     2319     2320 
## 54.40861 80.88678 80.82747 78.33113 80.59008 81.25141 79.23348 78.74442 
##     2321     2322     2323     2324     2325     2326     2327     2328 
## 78.52492 77.33655 76.63243 75.23947 74.91332 75.88382 74.69644 75.64449 
##     2329     2330     2331     2332     2333     2334     2335     2336 
## 75.49152 79.29961 77.62396 77.44581 77.30493 77.10142 77.30407 76.93436 
##     2337     2338     2339     2340     2341     2342     2343     2344 
## 76.64224 78.36844 75.99344 75.57970 75.32109 74.99037 74.65191 74.50929 
##     2345     2346     2347     2348     2349     2350     2351     2352 
## 72.59405 78.17417 80.49653 79.84020 80.71885 80.92762 80.66491 82.07443 
##     2353     2354     2355     2356     2357     2358     2359     2360 
## 79.37514 81.72167 81.30881 80.91154 78.54059 77.97128 78.01375 76.57676 
##     2361     2362     2363     2364     2365     2366     2367     2368 
## 76.43920 66.89612 67.36399 68.17010 68.07314 67.95112 66.92420 67.76765 
##     2369     2370     2371     2372     2373     2374     2375     2376 
## 66.53995 69.72607 70.09505 67.22795 66.82998 62.06520 66.22439 61.94317 
##     2377     2378     2379     2380     2381     2382     2383     2384 
## 62.20446 59.95647 60.43347 60.25911 59.77481 59.84630 59.46589 55.82502 
##     2385     2386     2387     2388     2389     2390     2391     2392 
## 58.47587 60.67799 51.64523 49.97989 51.82865 56.05160 56.42783 56.84529 
##     2393     2394     2395     2396     2397     2398     2399     2400 
## 56.74133 65.58428 63.64717 64.20860 68.37565 60.98972 59.45851 55.14164 
##     2401     2402     2403     2404     2405     2406     2407     2408 
## 52.57373 50.84075 50.35211 48.89529 48.71297 49.21261 50.69843 51.20503 
##     2409     2410     2411     2412     2413     2414     2415     2416 
## 52.92688 55.67836 55.86992 55.41743 55.20245 55.16905 48.24395 48.06013 
##     2417     2418     2419     2420     2421     2422     2423     2424 
## 48.58448 47.49094 47.59652 47.60569 47.38239 47.60908 47.81217 48.11315 
##     2425     2426     2427     2428     2429     2430     2431     2432 
## 53.93787 82.52165 81.03272 81.95283 81.91833 81.70466 79.94205 81.16648 
##     2433     2434     2435     2436     2437     2438     2439     2440 
## 82.12306 79.25478 80.30037 76.87396 79.68144 79.37098 78.43257 79.02414 
##     2441     2442     2443     2444     2445     2446     2447     2448 
## 78.89385 72.52322 72.22856 72.00265 71.84968 71.08488 71.76383 70.70478 
##     2449     2450     2451     2452     2453     2454     2455     2456 
## 70.76508 71.16634 71.06270 70.95474 69.68513 70.45908 70.31385 72.48122 
##     2457     2458     2459     2460     2461     2462     2463     2464 
## 69.18485 63.73325 64.04176 63.46046 63.32850 63.71394 60.31928 60.52366 
##     2465     2466     2467     2468     2469     2470     2471     2472 
## 60.06135 59.82493 63.23195 59.24950 57.95184 57.96570 57.89831 57.72771 
##     2473     2474     2475     2476     2477     2478     2479     2480 
## 57.63722 70.57437 70.46728 70.41168 70.11981 70.03225 74.60181 69.17582 
##     2481     2482     2483     2484     2485     2486     2487     2488 
## 68.62456 71.63040 68.10565 71.15546 60.63524 61.87819 61.79977 61.74028 
##     2489     2490     2491     2492     2493     2494     2495     2496 
## 62.45659 60.19330 62.37980 60.95863 58.29488 54.53670 50.99029 52.30735 
##     2497     2498     2499     2500     2501     2502     2503     2504 
## 41.37872 48.40851 37.35786 43.43470 42.38610 42.95453 33.07959 34.01486 
##     2505     2506     2507     2508     2509     2510     2511     2512 
## 35.86702 80.11885 79.99787 79.93751 82.34759 82.57008 79.63940 79.56918 
##     2513     2514     2515     2516     2517     2518     2519     2520 
## 81.88497 81.78430 81.46997 79.50746 82.37148 80.83163 80.40397 80.18380 
##     2521     2522     2523     2524     2525     2526     2527     2528 
## 80.19991 80.65055 84.18023 83.99871 83.77222 83.86323 82.94343 83.45571 
##     2529     2530     2531     2532     2533     2534     2535     2536 
## 83.02247 81.55397 81.11236 80.93966 80.69487 78.16261 80.02112 79.79853 
##     2537     2538     2539     2540     2541     2542     2543     2544 
## 78.36861 63.52493 63.76873 71.97135 71.60909 69.01045 69.22688 69.33573 
##     2545     2546     2547     2548     2549     2550     2551     2552 
## 67.54300 69.03841 68.90490 68.40108 67.83627 67.50350 67.35548 67.19553 
##     2553     2554     2555     2556     2557     2558     2559     2560 
## 67.05804 69.08716 70.30175 70.12383 69.99484 69.85896 69.72046 69.59889 
##     2561     2562     2563     2564     2565     2566     2567     2568 
## 66.75600 66.70124 66.45219 66.12749 66.23522 65.63735 65.13838 64.97119 
##     2569     2570     2571     2572     2573     2574     2575     2576 
## 64.76403 72.85263 72.75739 72.61757 71.56140 72.06727 71.50138 73.94524 
##     2577     2578     2579     2580     2581     2582     2583     2584 
## 71.13155 70.45037 70.34951 69.77624 69.35894 68.46437 67.82555 68.17848 
##     2585     2586     2587     2588     2589     2590     2591     2592 
## 67.81903 74.23137 76.04932 75.95469 75.91061 75.79500 75.88400 74.15665 
##     2593     2594     2595     2596     2597     2598     2599     2600 
## 73.35345 75.11158 73.00564 69.58618 69.57102 67.57540 67.48097 67.37977 
##     2601     2602     2603     2604     2605     2606     2607     2608 
## 67.33268 68.06542 68.05441 67.33192 67.31544 67.09612 66.95405 66.67213 
##     2609     2610     2611     2612     2613     2614     2615     2616 
## 66.00749 65.36757 64.66699 67.08436 63.07485 66.44557 62.20835 62.01065 
##     2617     2618     2619     2620     2621     2622     2623     2624 
## 50.65446 64.02925 63.92954 63.59847 63.30517 67.03857 60.82922 59.36445 
##     2625     2626     2627     2628     2629     2630     2631     2632 
## 58.50753 59.36051 59.00841 58.65210 58.52705 58.28753 58.10815 57.88698 
##     2633     2634     2635     2636     2637     2638     2639     2640 
## 57.94187 73.05529 73.16817 73.12254 72.95017 75.10877 72.91684 72.69016 
##     2641     2642     2643     2644     2645     2646     2647     2648 
## 72.85135 70.53076 70.51294 75.17374 74.04427 73.15324 71.51730 72.62894 
##     2649     2650     2651     2652     2653     2654     2655     2656 
## 73.78520 74.49255 73.29581 72.37381 72.97508 71.16858 71.67627 69.56814 
##     2657     2658     2659     2660     2661     2662     2663     2664 
## 70.28306 68.93246 69.66732 71.60926 73.63385 70.59509 69.66857 70.12388 
##     2665     2666     2667     2668     2669     2670     2671     2672 
## 67.39568 77.30632 75.21137 77.12651 76.98156 76.85283 76.52098 76.61810 
##     2673     2674     2675     2676     2677     2678     2679     2680 
## 76.26366 76.35079 76.17891 75.81085 75.45326 73.59774 74.94084 74.82915 
##     2681     2682     2683     2684     2685     2686     2687     2688 
## 72.70500 77.48136 77.78956 77.49844 75.28373 75.22099 73.91498 73.43927 
##     2689     2690     2691     2692     2693     2694     2695     2696 
## 71.31136 72.99459 70.15771 70.20617 69.74326 69.80621 69.08282 70.60646 
##     2697     2698     2699     2700     2701     2702     2703     2704 
## 68.18333 70.04001 69.89930 69.18138 73.05545 69.36864 64.41775 64.04114 
##     2705     2706     2707     2708     2709     2710     2711     2712 
## 63.65983 63.43496 63.47789 61.94620 62.20111 61.22899 63.46464 63.59638 
##     2713     2714     2715     2716     2717     2718     2719     2720 
## 55.51402 61.82786 61.17200 65.76322 60.07601 59.62423 59.04049 58.40697 
##     2721     2722     2723     2724     2725     2726     2727     2728 
## 57.76047 63.49138 63.02415 63.06425 56.05948 55.75721 62.42708 52.86447 
##     2729     2730     2731     2732     2733     2734     2735     2736 
## 51.96834 50.49794 72.41242 73.32636 70.20583 75.02511 75.00300 71.62780 
##     2737     2738     2739     2740     2741     2742     2743     2744 
## 71.24075 71.83388 72.18679 71.37298 71.72101 71.91472 71.78753 71.26057 
##     2745     2746     2747     2748     2749     2750     2751     2752 
## 70.48148 70.81644 75.71877 77.66175 73.66242 78.59751 75.43556 75.35182 
##     2753     2754     2755     2756     2757     2758     2759     2760 
## 75.27223 77.10225 76.73252 76.59464 76.23419 75.84386 74.10077 76.76732 
##     2761     2762     2763     2764     2765     2766     2767     2768 
## 75.19873 74.90035 82.29360 82.09664 82.72755 82.45962 81.61136 82.04951 
##     2769     2770     2771     2772     2773     2774     2775     2776 
## 82.25451 82.82571 84.06430 82.02273 78.57717 78.13549 80.83331 81.39110 
##     2777     2778     2779     2780     2781     2782     2783     2784 
## 79.70910 79.61191 65.00993 69.44566 62.93047 61.52239 58.97841 64.08974 
##     2785     2786     2787     2788     2789     2790     2791     2792 
## 55.12295 56.18969 53.69256 53.57208 49.23853 54.96892 50.37235 59.10981 
##     2793     2794     2795     2796     2797     2798     2799     2800 
## 56.58835 49.99795 83.33307 80.77121 81.35467 80.56111 82.51971 81.20597 
##     2801     2802     2803     2804     2805     2806     2807     2808 
## 81.57364 80.48049 81.51717 79.40264 78.23872 79.32635 78.95135 77.16557 
##     2809     2810     2811     2812     2813     2814     2815     2816 
## 74.23293 73.72361 77.66880 77.59992 77.57136 76.77118 77.44149 77.24814 
##     2817     2818     2819     2820     2821     2822     2823     2824 
## 75.05215 74.69525 76.12006 76.54463 76.37080 76.53247 75.82472 75.38663 
##     2825     2826     2827     2828     2829     2830     2831     2832 
## 75.24578 74.02310 71.43112 71.22383 71.06073 70.99733 70.93319 69.40162 
##     2833     2834     2835     2836     2837     2838     2839     2840 
## 69.29010 70.46429 70.24019 70.11961 69.87465 69.89464 69.58508 69.28826 
##     2841     2842     2843     2844     2845     2846     2847     2848 
## 68.99224 64.43630 70.71784 68.58311 68.43386 66.70706 68.26308 68.18473 
##     2849     2850     2851     2852     2853     2854     2855     2856 
## 67.93751 67.77504 67.63101 67.30822 63.00585 62.99216 62.57924 62.35977 
##     2857     2858     2859     2860     2861     2862     2863     2864 
## 62.19685 64.64516 72.59896 72.39142 72.38618 70.20801 72.05802 72.11060 
##     2865     2866     2867     2868     2869     2870     2871     2872 
## 71.87098 71.25507 70.74990 70.40186 69.73552 69.39352 69.17490 68.72663 
##     2873     2874     2875     2876     2877     2878     2879     2880 
## 68.15328 68.28720 70.93281 70.52026 69.55319 72.39850 69.99990 69.73805 
##     2881     2882     2883     2884     2885     2886     2887     2888 
## 69.46026 69.28154 69.09259 68.82034 68.56944 68.09888 67.83200 66.60234 
##     2889     2890     2891     2892     2893     2894     2895     2896 
## 67.63900 67.27937 63.45242 62.02831 63.34538 63.08494 62.71426 62.64522 
##     2897     2898     2899     2900     2901     2902     2903     2904 
## 62.45907 62.43100 66.25528 62.28339 62.09456 61.71411 61.54815 65.40294 
##     2905     2906     2907     2908     2909     2910     2911     2912 
## 61.26469 60.99697 68.10108 62.99530 62.36775 61.39485 60.14202 58.99516 
##     2913     2914     2915     2916     2917     2918     2919     2920 
## 60.15632 62.60483 53.46683 51.47682 50.29597 49.27673 57.84015 56.88280 
##     2921     2922     2923     2924     2925     2926     2927     2928 
## 47.02864 46.47385 59.73936 61.43472 59.69966 57.54467 55.82418 50.72720 
##     2929     2930     2931     2932     2933     2934     2935     2936 
## 48.43725 46.26522 54.59572 54.04709 39.56237 37.79854 36.67575 47.67924 
##     2937     2938 
## 35.44955 34.54925
actual = life_exp1$Life.expectancy
actual
##    [1] 65.0 59.9 59.9 59.5 59.2 58.8 58.6 58.1 57.5 57.3 57.3 57.0 56.7 56.2
##   [15] 55.3 54.8 77.8 77.5 77.2 76.9 76.6 76.2 76.1 75.3 75.9 74.2 73.5 73.0
##   [29] 72.8 73.3 73.6 72.6 75.6 75.4 75.3 75.1 74.9 74.7 74.4 74.1 73.8 73.4
##   [43] 72.9 72.3 71.7 71.6 71.4 71.3 52.4 51.7 51.1 56.0 51.0 49.6 49.1 48.7
##   [57] 48.2 47.7 47.4 47.1 46.8 46.5 45.7 45.3 76.4 76.2 76.1 75.9 75.7 75.6
##   [71] 75.4 75.2 75.0 74.8 74.6 74.4 74.2 74.0 73.8 73.6 76.3 76.2 76.0 75.9
##   [85] 75.7 75.5 75.6 75.4 74.8 75.2 74.9 74.7 74.1 74.1 74.0 74.1 74.8 74.6
##   [99] 74.4 74.4 73.9 73.5 73.3 73.2 73.5 72.9 73.0 73.0 72.7 72.6 72.6 72.0
##  [113] 82.8 82.7 82.5 82.3 82.0 81.9 81.7 81.3 81.3 81.2 81.0 86.0 83.0 79.9
##  [127] 79.9 79.5 81.5 81.4 81.1 88.0 88.0 84.0 82.0 84.0 81.0 79.8 79.4 79.3
##  [141] 78.8 78.7 78.6 78.1 72.7 72.5 72.2 71.9 71.6 71.1 78.0 73.0 73.0 69.2
##  [155] 68.4 68.4 67.8 67.8 67.5 66.6 76.1 75.4 74.8 74.9 75.0 75.0 74.6 74.5
##  [169] 74.4 74.2 74.1 73.8 73.2 73.1 72.9 72.6 76.9 76.8 76.7 76.5 76.1 76.1
##  [183] 76.0 75.8 75.6 75.5 75.3 75.2 75.0 74.9 74.7 74.5 71.8 71.4 71.0 77.0
##  [197] 73.0 69.9 69.5 69.1 68.6 68.2 67.8 67.3 66.8 66.3 65.8 65.3 75.5 75.4
##  [211] 75.2 75.1 74.9 74.7 74.6 74.4 74.2 74.1 73.9 73.8 73.7 73.5 73.4 73.3
##  [225] 72.3 72.0 71.7 71.9 72.0 73.0 70.0 70.0 69.8 68.9 68.1 68.2 67.7 67.2
##  [239] 67.7 68.0 81.1 89.0 87.0 83.0 83.0 80.0 79.8 79.5 79.5 79.4 78.9 78.8
##  [253] 78.3 78.0 78.0 77.6 71.0 70.0 69.8 69.4 69.4 69.5 69.5 69.6 69.6 69.4
##  [267] 69.0 68.7 68.4 68.5 68.2 68.3 60.0 59.7 59.5 59.3 59.1 58.7 58.4 57.6
##  [281] 57.1 56.8 56.5 56.1 55.8 55.6 55.5 55.4 69.8 69.4 69.1 68.7 68.3 67.9
##  [295] 67.4 67.0 66.5 65.8 65.0 64.2 63.3 62.5 61.7 62.0 77.0 74.0 71.0 69.8
##  [309] 69.3 68.7 68.0 67.4 66.8 66.2 65.7 65.1 64.5 63.9 63.3 62.6 77.4 77.2
##  [323] 77.0 76.8 76.9 76.4 76.1 76.0 75.4 75.7 75.0 75.5 75.2 75.4 74.9 74.6
##  [337] 65.7 65.1 64.2 63.4 62.2 61.1 59.2 57.5 56.9 54.8 51.7 48.1 46.4 46.0
##  [351] 46.7 47.8 75.0 74.8 74.7 74.5 74.1 73.8 73.6 73.4 73.3 73.0 72.7 72.0
##  [365] 71.8 71.4 71.0 75.0 77.7 77.6 77.1 78.3 77.4 76.9 76.8 77.2 76.0 76.3
##  [379] 76.2 76.4 76.0 74.8 74.7 74.4 74.5 74.3 74.1 73.9 73.7 73.4 73.2 72.9
##  [393] 72.6 72.2 72.1 72.2 72.0 71.8 71.6 71.1 59.9 59.3 59.0 58.6 58.1 57.5
##  [407] 56.9 56.1 55.3 54.3 53.3 52.4 51.6 51.0 56.0 51.0 59.6 59.1 58.6 58.0
##  [421] 57.4 56.8 56.2 55.3 54.8 54.1 53.4 52.6 51.9 51.5 51.3 58.0 53.3 52.8
##  [435] 52.3 52.0 51.7 51.5 51.0 54.0 49.9 49.4 48.7 48.2 48.0 47.7 47.8 47.9
##  [449] 73.3 73.0 72.8 72.7 72.6 72.5 72.4 72.4 72.3 72.1 71.8 71.4 71.1 77.0
##  [463] 73.0 69.9 68.7 68.3 67.8 67.4 67.0 66.6 66.1 65.6 65.0 64.1 62.9 61.5
##  [477] 63.0 59.3 58.5 57.7 57.3 56.7 56.4 55.9 55.6 55.3 54.8 54.2 53.6 53.3
##  [491] 52.8 52.1 51.8 51.6 51.5 51.4 82.2 82.0 81.8 81.6 81.5 81.2 81.0 87.0
##  [505] 85.0 85.0 81.0 80.0 79.7 79.5 79.4 79.1 52.5 58.0 49.9 53.0 49.8 49.2
##  [519] 48.6 47.6 46.8 46.3 45.9 45.7 45.7 45.6 45.6 46.0 53.1 52.6 52.2 51.8
##  [533] 51.6 51.2 57.0 49.6 49.4 48.5 48.6 48.5 48.4 48.1 48.0 47.6 85.0 83.0
##  [547] 81.0 79.9 79.8 79.1 79.3 79.6 78.9 78.9 78.4 78.0 77.9 77.8 77.3 77.3
##  [561] 76.1 75.8 75.6 75.4 75.2 75.0 74.9 74.5 74.4 74.2 73.9 73.5 73.1 72.7
##  [575] 72.2 71.7 74.8 74.6 74.4 74.3 74.2 73.6 73.6 73.5 73.5 73.1 73.1 72.8
##  [589] 72.4 71.8 71.5 71.4 63.5 63.2 62.9 62.5 62.2 61.8 61.3 61.0 66.0 63.0
##  [603] 60.0 59.8 59.6 59.5 59.5 59.5 64.7 64.2 63.9 63.7 62.9 62.0 68.0 59.4
##  [617] 58.2 56.9 55.3 54.1 53.2 52.6 52.7 52.9 73.4 79.6 79.5 79.4 79.2 79.0
##  [631] 78.1 79.2 78.9 78.9 78.0 78.6 77.7 78.0 78.3 77.5 77.6 78.0 77.8 77.7
##  [645] 77.1 77.0 76.6 76.3 76.0 75.8 75.9 75.2 75.4 74.7 74.8 74.9 74.7 79.1
##  [659] 79.0 78.7 78.7 78.8 78.0 78.1 77.9 78.1 78.0 77.2 77.3 77.4 77.7 76.7
##  [673] 76.9 85.0 83.0 81.0 80.0 79.7 79.5 79.3 79.1 78.9 78.8 78.7 78.6 78.5
##  [687] 78.4 78.2 78.1 78.8 78.6 78.2 78.0 77.8 77.5 77.1 77.0 76.8 76.5 75.9
##  [701] 75.8 75.2 75.3 75.1 74.7 76.0 73.0 71.0 69.8 69.4 69.0 68.7 68.6 68.5
##  [715] 68.5 68.5 68.4 68.1 67.6 66.6 65.4 59.8 59.3 58.8 58.3 57.9 57.4 56.7
##  [729] 56.3 55.7 55.0 54.3 53.5 52.8 52.1 51.8 51.3 86.0 84.0 81.0 80.0 79.7
##  [743] 79.2 78.9 78.8 78.4 78.1 78.1 77.7 77.3 77.0 77.0 76.9 63.5 63.0 62.7
##  [757] 62.2 61.8 61.3 69.0 62.0 59.8 59.1 58.6 58.1 58.0 57.9 57.7 57.4 73.6
##  [771] 73.9 73.6 73.4 72.1 73.1 72.7 73.6 73.3 72.9 72.3 69.7 69.3 73.0 71.4
##  [785] 71.2 72.0 76.2 76.0 76.0 75.5 75.3 75.0 75.1 74.6 74.7 74.4 74.2 74.4
##  [799] 74.4 73.6 73.4 72.8 79.0 78.0 79.0 72.0 74.0 70.0 69.9 69.8 69.7 69.5
##  [813] 69.4 69.0 68.6 68.7 68.6 68.8 73.5 73.3 73.0 73.0 72.0 72.0 71.4 71.7
##  [827] 71.2 75.0 71.0 70.0 69.9 73.0 68.9 69.0 58.2 57.9 57.4 56.7 56.2 56.1
##  [841] 55.7 55.4 55.0 54.8 54.4 54.1 53.8 53.5 53.1 52.7 64.7 64.4 64.0 63.6
##  [855] 62.9 62.1 61.4 67.0 62.0 59.7 59.4 59.1 58.8 58.5 58.1 45.3 77.6 77.3
##  [869] 76.9 76.3 76.1 75.6 74.9 74.2 73.0 73.0 72.8 72.3 71.9 71.2 78.0 78.0
##  [883] 64.8 64.2 63.7 63.3 62.6 61.8 68.0 59.8 58.5 57.2 56.0 55.0 54.0 53.2
##  [897] 52.5 51.2 69.9 69.7 69.6 69.4 69.2 69.1 68.9 68.7 68.6 68.5 68.3 68.1
##  [911] 68.0 67.9 67.8 67.7 81.1 89.0 87.0 84.0 83.0 79.9 79.7 79.6 79.3 79.2
##  [925] 78.9 78.7 78.4 78.1 78.0 77.5 82.4 82.2 82.0 81.5 81.7 81.3 81.1 89.0
##  [939] 89.0 86.0 81.0 82.0 79.3 79.2 79.0 78.8 66.0 65.5 64.6 63.5 62.8 62.3
##  [953] 61.7 61.6 61.6 61.4 65.0 59.7 59.7 59.7 59.8 61.0 61.1 68.0 66.0 62.0
##  [967] 59.8 59.3 59.0 58.7 58.5 58.2 57.7 57.3 57.0 56.6 56.3 55.9 74.4 74.5
##  [981] 74.5 74.2 73.9 73.8 73.2 73.9 74.4 73.9 73.9 72.3 72.7 71.7 73.0 71.8
##  [995] 81.0 89.0 86.0 86.0 85.0 81.0 80.0 79.9 79.8 79.6 79.2 79.1 78.5 78.4
## [1009] 78.3 78.0 62.4 62.1 61.9 61.6 61.2 69.0 66.0 63.0 59.9 59.4 58.9 58.3
## [1023] 57.9 57.6 57.4 57.2 81.0 88.0 86.0 84.0 85.0 83.0 80.0 79.9 79.4 79.7
## [1037] 79.3 79.2 79.1 79.0 78.7 78.2 73.6 73.5 73.3 73.1 72.9 72.6 72.4 72.1
## [1051] 71.9 71.7 71.5 73.0 71.1 79.0 77.0 74.0 71.9 71.7 71.4 71.3 71.1 77.0
## [1065] 76.0 79.0 75.0 69.7 69.2 69.6 69.4 69.3 68.4 67.7 59.0 58.1 58.8 58.4
## [1079] 58.1 57.8 57.3 56.8 56.4 55.6 54.7 54.0 53.3 52.9 52.5 52.5 58.9 58.4
## [1093] 58.1 57.6 57.1 56.7 56.3 55.6 55.0 54.4 53.9 53.5 53.0 52.8 52.5 52.1
## [1107] 66.2 66.0 65.9 65.8 65.6 65.9 66.1 66.3 65.7 65.2 65.0 65.1 65.3 65.3
## [1121] 65.4 65.4 63.5 63.1 62.7 62.3 62.3 36.3 62.5 62.1 61.8 61.1 65.0 58.7
## [1135] 59.7 59.3 58.9 58.6 74.6 74.5 74.3 74.1 73.9 73.6 73.4 73.2 73.0 72.8
## [1149] 72.5 72.2 71.9 71.6 71.3 71.0 75.8 75.6 75.5 75.0 74.8 74.5 74.2 74.1
## [1163] 73.5 73.4 72.9 72.9 72.5 72.5 72.3 71.7 82.7 82.5 82.4 82.5 82.1 81.8
## [1177] 81.6 81.4 81.3 81.1 81.0 88.0 87.0 84.0 80.0 79.7 68.3 68.0 67.6 67.3
## [1191] 66.8 66.4 66.0 65.5 65.2 64.8 64.4 64.0 63.7 63.3 62.9 62.5 69.1 68.9
## [1205] 68.7 68.5 68.3 68.1 67.9 67.7 67.5 67.3 67.2 65.3 66.9 66.7 66.5 66.3
## [1219] 75.5 75.4 75.3 75.1 74.7 74.1 73.3 72.7 72.4 72.2 72.0 71.8 75.0 71.2
## [1233] 78.0 73.0 68.9 67.9 69.5 76.0 77.0 76.0 74.0 69.3 65.9 64.7 66.8 67.2
## [1247] 66.5 74.0 72.0 70.0 81.4 81.2 81.0 85.0 84.0 86.0 79.7 79.8 79.5 79.0
## [1261] 78.7 78.3 78.0 77.4 77.0 76.4 82.5 82.2 82.1 81.8 81.8 81.7 81.5 81.0
## [1275] 84.0 84.0 80.0 81.0 79.7 79.3 79.3 78.9 82.7 82.5 82.3 82.0 82.0 81.8
## [1289] 81.6 81.5 81.3 81.2 88.0 89.0 79.9 80.0 79.8 79.4 76.2 75.8 75.6 75.3
## [1303] 75.2 75.0 74.7 74.5 74.2 74.0 73.5 73.3 73.1 73.0 72.7 72.6 83.7 83.5
## [1317] 83.5 83.3 82.5 83.0 83.0 82.7 82.6 82.4 82.0 82.1 81.9 81.8 81.5 81.1
## [1331] 74.1 74.0 73.9 73.7 73.6 73.4 73.3 73.1 73.0 72.8 72.4 72.5 72.3 72.1
## [1345] 71.9 71.7 72.0 69.9 69.5 69.1 68.5 67.8 67.8 66.6 65.3 65.0 64.6 64.7
## [1359] 64.4 64.7 64.4 63.9 63.4 62.9 62.6 62.1 61.2 63.0 59.1 57.9 56.8 55.3
## [1373] 54.1 53.0 52.4 52.1 51.9 51.9 66.3 66.1 65.8 65.7 65.5 65.3 65.2 65.1
## [1387] 65.0 65.0 64.9 64.8 64.7 64.6 64.3 64.1 74.7 74.6 74.5 74.3 74.2 74.0
## [1401] 73.9 73.8 73.7 73.6 73.6 73.5 73.4 73.3 73.2 73.2 71.1 78.0 77.0 69.9
## [1415] 69.4 68.8 68.5 67.6 67.2 66.7 66.9 67.1 66.6 66.7 67.2 66.6 65.7 65.3
## [1429] 64.9 64.4 64.0 63.6 63.1 62.6 62.1 61.5 61.0 64.0 59.8 59.3 58.7 58.1
## [1443] 74.6 74.4 74.1 73.8 73.6 72.8 72.6 71.9 78.0 75.0 76.0 71.0 78.0 73.0
## [1457] 69.9 71.0 74.9 74.8 74.9 75.0 75.0 74.9 74.7 74.5 74.4 74.1 73.9 73.7
## [1471] 73.5 73.2 73.0 72.7 53.7 52.1 52.1 52.2 52.3 51.1 49.4 47.8 46.2 45.3
## [1485] 44.5 44.8 45.5 46.4 47.8 49.3 61.4 58.1 61.1 67.0 62.0 59.7 59.2 58.6
## [1499] 57.9 56.7 55.3 54.0 50.0 56.0 51.5 51.9 72.7 72.4 72.9 72.9 71.3 72.8
## [1513] 72.7 72.6 72.5 72.2 71.9 71.5 71.3 71.1 71.0 78.0 73.6 73.4 73.0 73.0
## [1527] 72.8 72.4 72.2 71.1 72.0 76.0 78.0 71.6 71.6 71.4 71.2 71.6 82.0 81.7
## [1541] 81.4 81.1 88.0 86.0 83.0 80.0 79.7 79.4 78.8 78.7 78.6 78.3 78.0 77.8
## [1555] 65.5 65.1 64.7 64.3 63.8 63.3 62.8 62.3 61.9 61.4 69.0 64.0 59.9 59.3
## [1569] 58.7 57.9 58.3 57.6 56.7 55.3 54.1 52.9 51.5 50.0 48.5 47.1 46.0 45.1
## [1583] 44.6 44.0 43.5 43.1 75.0 74.8 74.6 74.5 74.3 74.1 74.0 73.8 73.7 73.6
## [1597] 73.4 73.2 73.1 72.9 72.7 72.4 78.5 78.2 77.9 77.6 77.3 76.7 76.3 75.9
## [1611] 75.4 75.0 74.3 73.4 72.7 71.8 78.0 69.6 58.2 57.8 57.3 57.2 56.8 56.5
## [1625] 56.0 55.5 55.0 54.3 53.6 52.8 52.0 51.2 55.0 49.8 81.7 81.4 81.1 81.0
## [1639] 87.0 83.0 82.0 80.0 79.6 79.3 79.0 78.7 78.5 78.2 77.8 77.5 72.2 63.1
## [1653] 63.0 62.7 62.5 62.2 62.0 61.7 61.4 61.2 69.0 66.0 64.0 63.0 62.0 61.0
## [1667] 60.0 74.6 74.2 74.1 73.9 73.6 73.3 72.8 72.7 72.9 71.8 72.1 71.9 71.5
## [1681] 71.5 71.5 71.0 76.7 76.6 76.6 76.3 76.1 75.6 75.7 75.6 76.0 75.8 75.3
## [1695] 75.4 75.0 75.0 75.0 74.8 69.4 69.4 69.2 69.0 68.9 68.7 68.5 68.4 68.2
## [1709] 68.0 67.9 67.7 67.5 66.2 67.2 67.0 77.0 68.8 68.4 68.1 67.8 67.3 66.3
## [1723] 66.9 67.4 65.9 65.0 64.5 64.0 64.0 63.8 63.2 62.8 76.1 75.9 75.8 75.6
## [1737] 75.4 75.3 75.0 74.6 74.2 73.8 73.6 73.5 73.5 73.4 73.3 73.0 74.3 74.1
## [1751] 73.9 73.6 73.3 72.8 72.3 71.8 71.4 71.0 77.0 72.0 69.9 69.5 69.0 68.6
## [1765] 57.6 56.7 55.3 54.8 54.3 54.0 53.8 53.2 52.1 51.2 58.0 54.0 51.0 49.8
## [1779] 49.5 49.0 66.6 66.4 66.2 65.9 65.6 65.4 65.2 59.2 64.5 64.2 63.9 63.5
## [1793] 63.2 62.8 62.5 62.1 65.8 65.9 66.1 65.8 64.3 63.0 62.4 61.7 60.0 57.0
## [1807] 55.1 54.7 55.0 55.7 56.5 57.4 72.8 69.2 69.6 69.3 68.9 68.4 68.0 67.5
## [1821] 67.0 66.6 66.0 65.4 64.7 64.3 63.1 63.2 62.5 81.9 81.7 81.4 81.1 81.1
## [1835] 88.0 86.0 83.0 82.0 79.8 79.4 79.2 78.7 78.4 78.3 78.1 81.6 81.5 81.3
## [1849] 81.1 86.0 89.0 85.0 81.0 81.0 79.9 79.9 79.2 79.1 78.7 78.5 78.6 74.8
## [1863] 74.5 73.9 73.9 74.5 73.2 73.2 72.5 72.5 73.0 71.2 71.0 76.0 75.0 73.0
## [1877] 73.0 61.8 61.4 69.0 63.0 59.4 58.2 57.1 56.0 55.2 54.5 53.7 52.9 52.1
## [1891] 51.4 56.0 50.0 54.5 53.6 53.2 52.7 52.3 52.0 51.6 59.0 55.0 49.8 49.2
## [1905] 48.5 48.1 47.7 47.4 47.1 77.0 81.8 81.6 81.5 81.3 81.1 81.0 89.0 86.0
## [1919] 85.0 84.0 81.0 79.8 79.4 78.9 78.8 78.5 76.6 76.4 76.2 76.0 75.8 75.6
## [1933] 75.4 75.2 74.9 74.7 74.3 74.0 73.6 73.3 72.9 72.6 66.4 66.2 66.0 65.7
## [1947] 65.5 65.1 64.8 64.6 64.4 64.2 62.9 63.7 63.5 63.2 63.0 62.8 79.0 77.8
## [1961] 77.6 77.5 77.2 77.3 76.5 76.8 76.5 76.4 76.2 75.8 75.8 75.5 75.7 75.5
## [1975] 75.7 62.9 62.7 62.4 62.2 62.0 61.8 61.6 61.4 61.1 68.0 64.0 59.9 59.6
## [1989] 59.3 59.1 58.9 74.0 73.9 73.8 73.6 73.4 73.2 73.0 72.7 72.5 72.3 72.1
## [2003] 71.9 71.7 71.5 71.2 79.0 75.5 75.3 75.3 74.9 74.5 73.7 73.8 73.9 74.0
## [2017] 74.2 72.8 72.2 72.1 72.6 72.4 71.4 68.5 68.4 68.1 68.1 68.0 67.9 68.0
## [2031] 67.5 67.5 67.3 67.0 67.3 67.2 66.8 66.8 66.8 77.5 77.3 77.1 76.8 76.7
## [2045] 76.3 75.7 75.5 75.3 75.2 75.0 74.9 74.7 74.5 74.2 73.7 81.1 89.0 86.0
## [2059] 83.0 82.0 79.6 79.3 79.0 78.7 78.5 77.7 78.0 77.3 77.2 76.9 76.6 78.2
## [2073] 78.1 77.9 77.8 77.5 77.3 77.0 76.8 76.7 76.6 76.6 76.6 76.5 76.4 76.3
## [2087] 76.2 82.3 82.0 81.7 81.2 81.1 87.0 86.0 83.0 79.8 79.4 78.7 78.2 77.6
## [2101] 77.1 76.7 76.0 72.1 71.8 71.7 79.0 77.0 68.8 69.0 68.9 68.3 68.0 67.3
## [2115] 68.0 67.6 67.5 67.6 67.1 75.0 74.8 74.6 74.4 74.3 73.4 73.1 73.1 72.9
## [2129] 72.5 71.9 71.7 71.1 77.0 78.0 77.0 75.0 73.0 70.0 69.6 69.4 68.4 68.2
## [2143] 67.5 67.3 66.4 65.0 64.9 64.6 64.8 65.1 65.0 66.1 65.7 65.2 64.6 63.8
## [2157] 62.8 61.0 68.0 59.6 57.6 55.3 53.4 52.0 57.0 48.6 48.3 76.8 75.2 75.0
## [2171] 74.8 74.7 74.6 74.2 74.3 74.1 73.9 73.5 73.1 72.6 72.2 72.0 71.8 71.6
## [2185] 73.2 73.1 72.7 72.8 72.7 72.5 72.3 72.1 71.9 71.7 71.4 71.2 71.0 79.0
## [2199] 79.0 79.0 74.0 73.8 73.6 73.2 73.0 72.6 76.0 72.5 72.2 72.0 71.6 71.4
## [2213] 79.0 76.0 75.0 72.0 78.6 67.5 67.3 67.1 66.9 66.6 66.2 65.8 65.4 65.1
## [2227] 64.7 64.3 63.8 63.4 63.1 62.8 62.6 74.5 74.4 74.3 74.1 73.9 73.7 73.4
## [2241] 73.3 73.2 73.2 73.1 73.1 73.0 72.9 72.8 72.6 66.7 66.4 66.0 65.6 64.9
## [2255] 64.3 63.5 62.8 62.1 61.3 65.0 59.7 59.0 58.4 57.9 57.5 75.6 75.4 75.3
## [2269] 74.9 74.6 74.4 74.1 74.0 73.8 73.6 73.0 73.0 73.0 72.9 73.1 72.6 73.2
## [2283] 73.0 72.9 72.7 72.6 72.4 72.3 72.2 72.2 72.2 72.2 72.1 72.1 72.1 72.0
## [2297] 71.8 51.0 48.1 54.0 49.7 48.9 48.1 47.1 46.2 45.3 44.3 43.3 42.3 41.5
## [2311] 48.0 41.0 39.0 83.1 82.9 82.7 82.5 82.2 82.0 81.7 81.4 81.1 87.0 82.0
## [2325] 79.7 79.3 79.0 78.7 78.3 76.7 76.4 76.1 75.8 75.6 75.1 75.0 74.7 74.4
## [2339] 74.3 74.0 74.1 73.8 73.7 73.3 73.0 88.0 87.0 85.0 82.0 79.8 79.5 79.1
## [2353] 78.9 78.3 78.1 77.5 77.2 76.5 76.6 76.2 76.0 69.2 68.8 68.8 68.7 68.5
## [2367] 68.3 68.1 68.0 67.6 67.6 67.4 67.1 66.8 66.5 66.2 65.8 55.0 54.3 54.2
## [2381] 53.1 53.1 52.4 52.2 51.9 51.5 51.5 51.6 51.2 51.1 58.0 57.0 55.0 62.9
## [2395] 62.0 69.0 59.2 58.9 58.0 56.5 55.3 54.5 54.0 53.8 53.7 54.0 54.9 56.0
## [2409] 57.3 57.3 56.6 56.4 56.0 55.4 55.0 54.3 53.6 53.1 52.5 51.9 51.4 58.0
## [2423] 52.0 49.6 48.9 82.8 82.6 82.4 82.0 82.1 81.9 81.6 81.3 89.0 88.0 81.0
## [2437] 81.0 79.4 79.5 79.4 79.1 74.9 74.7 74.6 74.5 74.5 74.5 71.8 72.3 73.7
## [2451] 73.8 74.2 69.1 73.9 73.7 72.7 71.5 64.1 63.8 63.5 63.2 62.7 62.5 62.0
## [2465] 61.8 61.4 61.0 67.0 59.7 59.6 59.4 58.9 58.6 71.6 71.4 71.2 71.3 76.0
## [2479] 73.0 70.0 69.8 69.5 69.3 68.9 68.3 68.0 67.9 67.7 67.4 58.9 58.4 57.6
## [2493] 56.5 55.0 53.6 52.6 51.4 50.0 47.8 46.0 45.6 45.9 46.4 47.1 48.4 82.4
## [2507] 82.3 81.9 81.7 81.7 81.5 81.4 81.1 89.0 88.0 85.0 83.0 82.0 79.9 79.8
## [2521] 79.6 83.4 83.2 83.0 82.7 82.6 82.3 82.1 82.0 81.7 81.5 81.1 81.0 85.0
## [2535] 84.0 82.0 79.7 64.5 64.4 63.6 62.8 71.7 73.7 73.8 73.8 73.8 73.7 73.5
## [2549] 73.2 73.0 72.8 72.7 72.6 69.7 69.6 69.3 68.8 68.1 67.3 66.7 66.4 66.1
## [2563] 65.9 65.5 65.9 65.2 64.3 64.0 63.7 74.9 74.6 74.5 74.3 74.1 73.9 73.7
## [2577] 73.5 73.3 73.0 72.5 71.6 71.7 71.4 71.2 71.1 75.7 75.5 75.3 75.1 74.9
## [2591] 74.7 74.4 74.2 73.5 73.7 73.6 73.5 73.2 72.8 73.1 72.6 68.3 68.0 67.7
## [2605] 67.4 67.2 66.9 66.6 66.2 65.8 64.9 63.7 62.3 61.0 62.0 59.4 58.7 59.9
## [2619] 59.7 59.4 58.9 58.3 57.4 56.7 56.2 55.9 55.7 55.0 54.9 54.7 54.7 54.6
## [2633] 54.6 73.5 73.3 73.2 73.0 72.9 72.8 72.5 72.6 72.5 72.4 72.3 72.2 72.0
## [2647] 71.9 71.8 71.6 71.2 71.1 71.0 78.0 76.0 74.0 71.0 69.9 69.7 69.6 69.5
## [2661] 69.4 69.3 69.2 69.1 69.1 75.3 75.1 74.9 74.9 74.8 74.8 74.7 74.7 74.6
## [2675] 74.4 74.2 74.0 73.7 73.5 73.2 72.9 75.8 75.5 75.2 74.8 74.5 74.2 73.9
## [2689] 73.5 73.2 72.8 72.4 72.0 71.6 71.2 78.0 74.0 66.3 66.0 65.4 65.6 65.6
## [2703] 65.8 65.6 64.5 64.1 63.7 63.3 63.5 63.4 63.3 64.0 63.8 71.9 62.3 61.5
## [2717] 67.0 60.0 59.3 58.4 57.5 56.3 55.5 54.9 53.2 51.3 51.0 48.8 47.7 46.6
## [2731] 71.3 78.0 71.0 77.0 75.0 69.8 69.2 67.7 67.5 67.7 67.0 67.4 67.6 67.6
## [2745] 67.7 67.5 77.1 76.9 76.7 76.5 76.3 76.2 76.0 75.8 75.6 75.4 75.3 75.1
## [2759] 74.9 74.7 74.5 74.2 81.2 81.0 87.0 86.0 86.0 82.0 81.0 79.6 79.5 79.3
## [2773] 79.0 78.8 78.3 78.2 78.0 77.8 61.8 67.0 59.7 58.6 58.3 57.5 56.9 56.2
## [2787] 54.5 53.1 52.2 51.5 58.0 52.0 49.6 49.2 79.3 79.1 78.9 78.8 78.7 78.7
## [2801] 78.5 78.2 78.1 77.8 77.5 77.5 77.2 77.0 76.9 76.8 77.0 76.8 76.8 76.5
## [2815] 77.0 76.3 76.6 76.4 75.4 76.2 75.7 75.4 75.4 75.4 75.2 75.1 69.4 69.2
## [2829] 69.1 68.8 68.5 68.3 68.0 67.9 67.8 67.6 67.3 67.8 67.2 67.1 67.4 67.1
## [2843] 72.0 71.7 71.6 71.4 71.2 71.0 78.0 75.0 73.0 71.0 69.9 69.6 69.4 69.3
## [2857] 69.1 69.0 74.1 73.9 73.8 73.7 73.8 73.7 73.6 73.2 73.4 73.6 73.6 73.3
## [2871] 72.4 73.1 72.5 72.5 76.0 75.9 75.7 75.6 75.4 75.2 75.0 74.9 74.7 74.6
## [2885] 74.4 74.2 74.0 73.8 73.6 73.4 65.7 65.4 65.4 64.7 64.6 64.4 64.1 63.8
## [2899] 63.4 63.0 62.6 62.2 61.9 61.5 61.1 68.0 61.8 61.1 63.0 59.2 58.2 58.0
## [2913] 57.4 55.7 52.6 58.0 49.3 47.9 46.4 45.5 44.6 43.8 67.0 59.2 58.0 56.6
## [2927] 54.9 52.4 50.0 48.2 46.6 45.4 44.6 44.3 44.5 44.8 45.3 46.0
Backtrack=data.frame(actual, predicted)
Backtrack
##      actual predicted
## 1      65.0  62.53524
## 2      59.9  62.26711
## 3      59.9  62.16433
## 4      59.5  61.90735
## 5      59.2  61.44841
## 6      58.8  61.17593
## 7      58.6  60.75759
## 8      58.1  60.41790
## 9      57.5  59.79887
## 10     57.3  59.47177
## 11     57.3  59.36032
## 12     57.0  58.35034
## 13     56.7  57.98504
## 14     56.2  62.59840
## 15     55.3  57.91175
## 16     54.8  57.52762
## 17     77.8  76.20788
## 18     77.5  77.37115
## 19     77.2  75.95013
## 20     76.9  75.82490
## 21     76.6  74.96086
## 22     76.2  73.94551
## 23     76.1  73.84116
## 24     75.3  75.01618
## 25     75.9  74.46019
## 26     74.2  70.94189
## 27     73.5  73.50867
## 28     73.0  73.60120
## 29     72.8  73.21808
## 30     73.3  73.26021
## 31     73.6  73.07832
## 32     72.6  73.12305
## 33     75.6  76.84333
## 34     75.4  76.90202
## 35     75.3  75.26686
## 36     75.1  75.12568
## 37     74.9  74.60659
## 38     74.7  74.06946
## 39     74.4  73.47527
## 40     74.1  72.70952
## 41     73.8  70.84526
## 42     73.4  72.29059
## 43     72.9  69.36420
## 44     72.3  71.32731
## 45     71.7  68.36805
## 46     71.6  68.22164
## 47     71.4  68.02480
## 48     71.3  67.80198
## 49     52.4  62.67439
## 50     51.7  61.78257
## 51     51.1  61.56393
## 52     56.0  60.57957
## 53     51.0  59.79209
## 54     49.6  58.63253
## 55     49.1  58.73899
## 56     48.7  58.26489
## 57     48.2  57.66446
## 58     47.7  57.00412
## 59     47.4  57.17968
## 60     47.1  56.70129
## 61     46.8  56.21848
## 62     46.5  55.59039
## 63     45.7  61.58557
## 64     45.3  61.07044
## 65     76.4  75.72930
## 66     76.2  74.91354
## 67     76.1  74.80593
## 68     75.9  74.68122
## 69     75.7  74.91998
## 70     75.6  74.87481
## 71     75.4  76.61278
## 72     75.2  74.45076
## 73     75.0  75.02599
## 74     74.8  75.01374
## 75     74.6  57.57648
## 76     74.4  55.71881
## 77     74.2  56.07674
## 78     74.0  57.29806
## 79     73.8  57.26066
## 80     73.6  57.19919
## 81     76.3  78.99514
## 82     76.2  78.82922
## 83     76.0  78.82007
## 84     75.9  80.62966
## 85     75.7  78.54490
## 86     75.5  77.68838
## 87     75.6  77.66873
## 88     75.4  77.46305
## 89     74.8  77.25752
## 90     75.2  77.01992
## 91     74.9  76.89438
## 92     74.7  79.04264
## 93     74.1  76.72813
## 94     74.1  76.76732
## 95     74.0  73.78286
## 96     74.1  73.58054
## 97     74.8  73.71721
## 98     74.6  75.68134
## 99     74.4  73.64809
## 100    74.4  73.62446
## 101    73.9  73.40452
## 102    73.5  72.95570
## 103    73.3  70.92440
## 104    73.2  72.28333
## 105    73.5  69.97095
## 106    72.9  69.25228
## 107    73.0  70.37677
## 108    73.0  71.28447
## 109    72.7  71.06114
## 110    72.6  70.97759
## 111    72.6  71.08924
## 112    72.0  71.11517
## 113    82.8  86.48200
## 114    82.7  87.57512
## 115    82.5  86.74027
## 116    82.3  86.53648
## 117    82.0  85.95242
## 118    81.9  85.17850
## 119    81.7  84.37249
## 120    81.3  84.62017
## 121    81.3  82.34892
## 122    81.2  84.82019
## 123    81.0  83.17514
## 124    86.0  81.46071
## 125    83.0  82.24857
## 126    79.9  82.62845
## 127    79.9  83.77267
## 128    79.5  80.45117
## 129    81.5  81.33991
## 130    81.4  81.80753
## 131    81.1  79.12155
## 132    88.0  82.46191
## 133    88.0  78.55990
## 134    84.0  78.47308
## 135    82.0  78.37476
## 136    84.0  77.84930
## 137    81.0  78.85891
## 138    79.8  75.12090
## 139    79.4  76.73171
## 140    79.3  76.41356
## 141    78.8  74.71749
## 142    78.7  78.38163
## 143    78.6  74.60178
## 144    78.1  74.43005
## 145    72.7  73.67048
## 146    72.5  73.62130
## 147    72.2  71.58070
## 148    71.9  72.10910
## 149    71.6  71.92615
## 150    71.1  72.37954
## 151    78.0  69.85783
## 152    73.0  69.77200
## 153    73.0  71.88268
## 154    69.2  68.44323
## 155    68.4  68.64389
## 156    68.4  68.49397
## 157    67.8  68.19152
## 158    67.8  67.93367
## 159    67.5  67.64084
## 160    66.6  69.78579
## 161    76.1  74.54986
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## 1805   60.0  55.99926
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## 1811   56.5  50.35325
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## 1813   72.8  70.33218
## 1814   69.2  67.29795
## 1815   69.6  69.01703
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## 1829   62.5  61.05443
## 1830   81.9  83.70997
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## 1832   81.4  85.00037
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## 1843   78.4  80.93930
## 1844   78.3  80.57007
## 1845   78.1  80.24093
## 1846   81.6  82.77704
## 1847   81.5  82.72161
## 1848   81.3  84.58149
## 1849   81.1  83.94614
## 1850   86.0  84.52782
## 1851   89.0  84.76437
## 1852   85.0  82.29035
## 1853   81.0  81.04128
## 1854   81.0  79.74146
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## 1857   79.2  78.88789
## 1858   79.1  77.58946
## 1859   78.7  76.70912
## 1860   78.5  78.37101
## 1861   78.6  78.07018
## 1862   74.8  71.86798
## 1863   74.5  71.77059
## 1864   73.9  71.40159
## 1865   73.9  71.33458
## 1866   74.5  69.70102
## 1867   73.2  69.42269
## 1868   73.2  70.74493
## 1869   72.5  70.41802
## 1870   72.5  70.17886
## 1871   73.0  70.41786
## 1872   71.2  67.23972
## 1873   71.0  67.12359
## 1874   76.0  66.98915
## 1875   75.0  66.61994
## 1876   73.0  66.36121
## 1877   73.0  66.06380
## 1878   61.8  62.25878
## 1879   61.4  58.66182
## 1880   69.0  59.14555
## 1881   63.0  58.76839
## 1882   59.4  58.30225
## 1883   58.2  57.65441
## 1884   57.1  56.43583
## 1885   56.0  55.90588
## 1886   55.2  55.58926
## 1887   54.5  55.38958
## 1888   53.7  55.01107
## 1889   52.9  53.42523
## 1890   52.1  57.90795
## 1891   51.4  53.68127
## 1892   56.0  53.05907
## 1893   50.0  53.67798
## 1894   54.5  60.05219
## 1895   53.6  60.31809
## 1896   53.2  59.10012
## 1897   52.7  59.40369
## 1898   52.3  58.19823
## 1899   52.0  58.16072
## 1900   51.6  57.90603
## 1901   59.0  57.39400
## 1902   55.0  56.60705
## 1903   49.8  56.81607
## 1904   49.2  61.55346
## 1905   48.5  61.83455
## 1906   48.1  56.19488
## 1907   47.7  57.50883
## 1908   47.4  55.47835
## 1909   47.1  54.43451
## 1910   77.0  75.55101
## 1911   81.8  82.00653
## 1912   81.6  79.45623
## 1913   81.5  79.23893
## 1914   81.3  81.76145
## 1915   81.1  81.14395
## 1916   81.0  85.10189
## 1917   89.0  80.91431
## 1918   86.0  81.42248
## 1919   85.0  85.01924
## 1920   84.0  85.42600
## 1921   81.0  83.96663
## 1922   79.8  80.85756
## 1923   79.4  80.44941
## 1924   78.9  80.30771
## 1925   78.8  82.18010
## 1926   78.5  80.78887
## 1927   76.6  75.54209
## 1928   76.4  76.41022
## 1929   76.2  77.23605
## 1930   76.0  77.21864
## 1931   75.8  76.19820
## 1932   75.6  76.05683
## 1933   75.4  75.75244
## 1934   75.2  74.46674
## 1935   74.9  70.71020
## 1936   74.7  70.68491
## 1937   74.3  72.66134
## 1938   74.0  71.98177
## 1939   73.6  71.56340
## 1940   73.3  73.70976
## 1941   72.9  71.23121
## 1942   72.6  65.25482
## 1943   66.4  63.15032
## 1944   66.2  62.79318
## 1945   66.0  62.48705
## 1946   65.7  62.28178
## 1947   65.5  63.46560
## 1948   65.1  63.25874
## 1949   64.8  63.00474
## 1950   64.6  62.28478
## 1951   64.4  61.68213
## 1952   64.2  60.76199
## 1953   62.9  63.51657
## 1954   63.7  59.72414
## 1955   63.5  59.57795
## 1956   63.2  59.39208
## 1957   63.0  59.16499
## 1958   62.8  62.11241
## 1959   79.0  78.20699
## 1960   77.8  72.76947
## 1961   77.6  72.70416
## 1962   77.5  72.54950
## 1963   77.2  73.70515
## 1964   77.3  73.16806
## 1965   76.5  74.33424
## 1966   76.8  71.79010
## 1967   76.5  71.66359
## 1968   76.4  71.33360
## 1969   76.2  73.03325
## 1970   75.8  70.87939
## 1971   75.8  73.77135
## 1972   75.5  73.61317
## 1973   75.7  73.39673
## 1974   75.5  73.03655
## 1975   75.7  72.70858
## 1976   62.9  64.31285
## 1977   62.7  64.17437
## 1978   62.4  63.98624
## 1979   62.2  63.68206
## 1980   62.0  63.60210
## 1981   61.8  63.21519
## 1982   61.6  62.78015
## 1983   61.4  66.98840
## 1984   61.1  61.85830
## 1985   68.0  61.41722
## 1986   64.0  65.72762
## 1987   59.9  59.28729
## 1988   59.6  60.00025
## 1989   59.3  59.37853
## 1990   59.1  59.25520
## 1991   58.9  58.71018
## 1992   74.0  69.26677
## 1993   73.9  70.33276
## 1994   73.8  69.77801
## 1995   73.6  69.76240
## 1996   73.4  72.49038
## 1997   73.2  69.96654
## 1998   73.0  69.61317
## 1999   72.7  69.14092
## 2000   72.5  71.49164
## 2001   72.3  71.35263
## 2002   72.1  71.43874
## 2003   71.9  71.11687
## 2004   71.7  69.32220
## 2005   71.5  67.52400
## 2006   71.2  71.17047
## 2007   79.0  68.26475
## 2008   75.5  71.92409
## 2009   75.3  72.11275
## 2010   75.3  72.04941
## 2011   74.9  74.24509
## 2012   74.5  74.02791
## 2013   73.7  75.71472
## 2014   73.8  75.79712
## 2015   73.9  71.59029
## 2016   74.0  71.77371
## 2017   74.2  73.11153
## 2018   72.8  70.46929
## 2019   72.2  70.15816
## 2020   72.1  72.75528
## 2021   72.6  73.35551
## 2022   72.4  70.60083
## 2023   71.4  72.75613
## 2024   68.5  67.51778
## 2025   68.4  66.34069
## 2026   68.1  67.11953
## 2027   68.1  67.14911
## 2028   68.0  66.93936
## 2029   67.9  66.78702
## 2030   68.0  67.05874
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## 2525   82.7  83.77222
## 2526   82.6  83.86323
## 2527   82.3  82.94343
## 2528   82.1  83.45571
## 2529   82.0  83.02247
## 2530   81.7  81.55397
## 2531   81.5  81.11236
## 2532   81.1  80.93966
## 2533   81.0  80.69487
## 2534   85.0  78.16261
## 2535   84.0  80.02112
## 2536   82.0  79.79853
## 2537   79.7  78.36861
## 2538   64.5  63.52493
## 2539   64.4  63.76873
## 2540   63.6  71.97135
## 2541   62.8  71.60909
## 2542   71.7  69.01045
## 2543   73.7  69.22688
## 2544   73.8  69.33573
## 2545   73.8  67.54300
## 2546   73.8  69.03841
## 2547   73.7  68.90490
## 2548   73.5  68.40108
## 2549   73.2  67.83627
## 2550   73.0  67.50350
## 2551   72.8  67.35548
## 2552   72.7  67.19553
## 2553   72.6  67.05804
## 2554   69.7  69.08716
## 2555   69.6  70.30175
## 2556   69.3  70.12383
## 2557   68.8  69.99484
## 2558   68.1  69.85896
## 2559   67.3  69.72046
## 2560   66.7  69.59889
## 2561   66.4  66.75600
## 2562   66.1  66.70124
## 2563   65.9  66.45219
## 2564   65.5  66.12749
## 2565   65.9  66.23522
## 2566   65.2  65.63735
## 2567   64.3  65.13838
## 2568   64.0  64.97119
## 2569   63.7  64.76403
## 2570   74.9  72.85263
## 2571   74.6  72.75739
## 2572   74.5  72.61757
## 2573   74.3  71.56140
## 2574   74.1  72.06727
## 2575   73.9  71.50138
## 2576   73.7  73.94524
## 2577   73.5  71.13155
## 2578   73.3  70.45037
## 2579   73.0  70.34951
## 2580   72.5  69.77624
## 2581   71.6  69.35894
## 2582   71.7  68.46437
## 2583   71.4  67.82555
## 2584   71.2  68.17848
## 2585   71.1  67.81903
## 2586   75.7  74.23137
## 2587   75.5  76.04932
## 2588   75.3  75.95469
## 2589   75.1  75.91061
## 2590   74.9  75.79500
## 2591   74.7  75.88400
## 2592   74.4  74.15665
## 2593   74.2  73.35345
## 2594   73.5  75.11158
## 2595   73.7  73.00564
## 2596   73.6  69.58618
## 2597   73.5  69.57102
## 2598   73.2  67.57540
## 2599   72.8  67.48097
## 2600   73.1  67.37977
## 2601   72.6  67.33268
## 2602   68.3  68.06542
## 2603   68.0  68.05441
## 2604   67.7  67.33192
## 2605   67.4  67.31544
## 2606   67.2  67.09612
## 2607   66.9  66.95405
## 2608   66.6  66.67213
## 2609   66.2  66.00749
## 2610   65.8  65.36757
## 2611   64.9  64.66699
## 2612   63.7  67.08436
## 2613   62.3  63.07485
## 2614   61.0  66.44557
## 2615   62.0  62.20835
## 2616   59.4  62.01065
## 2617   58.7  50.65446
## 2618   59.9  64.02925
## 2619   59.7  63.92954
## 2620   59.4  63.59847
## 2621   58.9  63.30517
## 2622   58.3  67.03857
## 2623   57.4  60.82922
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## 2625   56.2  58.50753
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## 2630   54.7  58.28753
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## 2632   54.6  57.88698
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## 2634   73.5  73.05529
## 2635   73.3  73.16817
## 2636   73.2  73.12254
## 2637   73.0  72.95017
## 2638   72.9  75.10877
## 2639   72.8  72.91684
## 2640   72.5  72.69016
## 2641   72.6  72.85135
## 2642   72.5  70.53076
## 2643   72.4  70.51294
## 2644   72.3  75.17374
## 2645   72.2  74.04427
## 2646   72.0  73.15324
## 2647   71.9  71.51730
## 2648   71.8  72.62894
## 2649   71.6  73.78520
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## 2651   71.1  73.29581
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## 2655   74.0  71.67627
## 2656   71.0  69.56814
## 2657   69.9  70.28306
## 2658   69.7  68.93246
## 2659   69.6  69.66732
## 2660   69.5  71.60926
## 2661   69.4  73.63385
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## 2665   69.1  67.39568
## 2666   75.3  77.30632
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## 2691   72.8  70.15771
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## 2695   71.2  69.08282
## 2696   78.0  70.60646
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## 2698   66.3  70.04001
## 2699   66.0  69.89930
## 2700   65.4  69.18138
## 2701   65.6  73.05545
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## 2705   64.5  63.65983
## 2706   64.1  63.43496
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## 2708   63.3  61.94620
## 2709   63.5  62.20111
## 2710   63.4  61.22899
## 2711   63.3  63.46464
## 2712   64.0  63.59638
## 2713   63.8  55.51402
## 2714   71.9  61.82786
## 2715   62.3  61.17200
## 2716   61.5  65.76322
## 2717   67.0  60.07601
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## 2719   59.3  59.04049
## 2720   58.4  58.40697
## 2721   57.5  57.76047
## 2722   56.3  63.49138
## 2723   55.5  63.02415
## 2724   54.9  63.06425
## 2725   53.2  56.05948
## 2726   51.3  55.75721
## 2727   51.0  62.42708
## 2728   48.8  52.86447
## 2729   47.7  51.96834
## 2730   46.6  50.49794
## 2731   71.3  72.41242
## 2732   78.0  73.32636
## 2733   71.0  70.20583
## 2734   77.0  75.02511
## 2735   75.0  75.00300
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## 2778   77.8  79.61191
## 2779   61.8  65.00993
## 2780   67.0  69.44566
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## 2782   58.6  61.52239
## 2783   58.3  58.97841
## 2784   57.5  64.08974
## 2785   56.9  55.12295
## 2786   56.2  56.18969
## 2787   54.5  53.69256
## 2788   53.1  53.57208
## 2789   52.2  49.23853
## 2790   51.5  54.96892
## 2791   58.0  50.37235
## 2792   52.0  59.10981
## 2793   49.6  56.58835
## 2794   49.2  49.99795
## 2795   79.3  83.33307
## 2796   79.1  80.77121
## 2797   78.9  81.35467
## 2798   78.8  80.56111
## 2799   78.7  82.51971
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## 2801   78.5  81.57364
## 2802   78.2  80.48049
## 2803   78.1  81.51717
## 2804   77.8  79.40264
## 2805   77.5  78.23872
## 2806   77.5  79.32635
## 2807   77.2  78.95135
## 2808   77.0  77.16557
## 2809   76.9  74.23293
## 2810   76.8  73.72361
## 2811   77.0  77.66880
## 2812   76.8  77.59992
## 2813   76.8  77.57136
## 2814   76.5  76.77118
## 2815   77.0  77.44149
## 2816   76.3  77.24814
## 2817   76.6  75.05215
## 2818   76.4  74.69525
## 2819   75.4  76.12006
## 2820   76.2  76.54463
## 2821   75.7  76.37080
## 2822   75.4  76.53247
## 2823   75.4  75.82472
## 2824   75.4  75.38663
## 2825   75.2  75.24578
## 2826   75.1  74.02310
## 2827   69.4  71.43112
## 2828   69.2  71.22383
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## 2839   67.2  69.58508
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## 2859   74.1  72.59896
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## 2863   73.8  72.05802
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## 2875   76.0  70.93281
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## 2879   75.4  69.99990
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## 2885   74.4  68.56944
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## 2888   73.8  66.60234
## 2889   73.6  67.63900
## 2890   73.4  67.27937
## 2891   65.7  63.45242
## 2892   65.4  62.02831
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## 2894   64.7  63.08494
## 2895   64.6  62.71426
## 2896   64.4  62.64522
## 2897   64.1  62.45907
## 2898   63.8  62.43100
## 2899   63.4  66.25528
## 2900   63.0  62.28339
## 2901   62.6  62.09456
## 2902   62.2  61.71411
## 2903   61.9  61.54815
## 2904   61.5  65.40294
## 2905   61.1  61.26469
## 2906   68.0  60.99697
## 2907   61.8  68.10108
## 2908   61.1  62.99530
## 2909   63.0  62.36775
## 2910   59.2  61.39485
## 2911   58.2  60.14202
## 2912   58.0  58.99516
## 2913   57.4  60.15632
## 2914   55.7  62.60483
## 2915   52.6  53.46683
## 2916   58.0  51.47682
## 2917   49.3  50.29597
## 2918   47.9  49.27673
## 2919   46.4  57.84015
## 2920   45.5  56.88280
## 2921   44.6  47.02864
## 2922   43.8  46.47385
## 2923   67.0  59.73936
## 2924   59.2  61.43472
## 2925   58.0  59.69966
## 2926   56.6  57.54467
## 2927   54.9  55.82418
## 2928   52.4  50.72720
## 2929   50.0  48.43725
## 2930   48.2  46.26522
## 2931   46.6  54.59572
## 2932   45.4  54.04709
## 2933   44.6  39.56237
## 2934   44.3  37.79854
## 2935   44.5  36.67575
## 2936   44.8  47.67924
## 2937   45.3  35.44955
## 2938   46.0  34.54925
library(Metrics)
data.frame(Method = c("MSE","RMSE","MAE", "MAPE"), 
           Error.Value = c(mse(predicted, actual),
                          rmse(predicted, actual),
                          mae(predicted, actual),
                          mape(predicted, actual)))
##   Method Error.Value
## 1    MSE 15.95766123
## 2   RMSE  3.99470415
## 3    MAE  2.97040801
## 4   MAPE  0.04505506
range(actual)
## [1] 36.3 89.0

When compared to the Life.expectancy as the dependent variable’s range, the error values from each technique appear to be tiny. We may therefore anticipate that the predicted values will be fairly close to the actual ones.

plot(actual,col="Red")
lines(actual,col="Red")

plot(predicted,col="Blue")
lines(predicted,col="Blue")
lines(actual,col="Red")

From the above chart, we can interpret that the predicted values are close enough to the actual. Therefore, the model is a good fit.

Checking on Assumptions

Normality Test

Ploting Residuals on Histogram
hist(model_backward$residuals, breaks = 20)

The majority of the residuals appear to be distributed in the middle, suggesting that their distribution is normal. ##### Ploting Residuals on QQPlot

plot(model_backward, which = 2)

The majority of the residuals collected at the middle line suggest that their distribution is normal.

Shapiro Test
shapiro.test(model_regs$residuals)
## 
##  Shapiro-Wilk normality test
## 
## data:  model_regs$residuals
## W = 0.96622, p-value < 2.2e-16

Based on Shapiro-Wilk normality test, the p-value < 0.05 implying that the distribution of the data are significantly different from normal distribution. Therefore, we need to do some adjustment to data.

Remove Outliers
boxplot(life_exp1$Life.expectancy, ylab = "Life Expectancy (Age)") # visual boxplot

outliers_out <- boxplot(life_exp1$Life.expectancy, plot = F)$out # untuk mendaptkan outlier
life_clean <- life_exp1[-which(life_exp1$Life.expectancy %in% outliers_out), ] # remove outlier dari data

Let us see the Boxplot after Outliers taken

boxplot(life_clean$Life.expectancy, ylab = "Life Expectancy (Age)") # visual boxplot

Unfortunately, there are still some Outliers, so we will eliminate all data with Life.expectancy 50 and below.

life_clean1 <- life_clean[life_clean$Life.expectancy > 50, ] # Eliminate all below Age 50
boxplot(life_clean1$Life.expectancy, ylab = "Life Expectancy (Age)")

##### Creating New Model

clean_full <- lm(formula = Life.expectancy ~., data = life_clean1)
clean_none <- lm(formula = Life.expectancy ~1, data = life_clean1)

clean_backward <- step(clean_full, direction = "backward")
## Start:  AIC=7257.33
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Hepatitis.B_above90 + 
##     Polio_below90 + Polio_above90 + Diphtheria_below90 + Diphtheria_above90
## 
## 
## Step:  AIC=7257.33
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Hepatitis.B_above90 + 
##     Polio_below90 + Polio_above90 + Diphtheria_below90
## 
## 
## Step:  AIC=7257.33
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Hepatitis.B_above90 + 
##     Polio_below90 + Diphtheria_below90
## 
## 
## Step:  AIC=7257.33
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Developing + Hepatitis.B_below90 + Polio_below90 + 
##     Diphtheria_below90
## 
## 
## Step:  AIC=7257.33
## Life.expectancy ~ Adult.Mortality + infant.deaths + Measles + 
##     BMI + Total.expenditure + HIV.AIDS + GDP + Population + thinness..1.19.years + 
##     thinness.5.9.years + Income.composition.of.resources + Schooling + 
##     Developed + Hepatitis.B_below90 + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - Measles                          1       6.0 36722 7255.8
## - Population                       1       6.8 36723 7255.9
## - infant.deaths                    1       9.4 36725 7256.1
## - thinness.5.9.years               1      10.9 36727 7256.2
## <none>                                         36716 7257.3
## - Hepatitis.B_below90              1      30.8 36747 7257.7
## - Diphtheria_below90               1      98.5 36814 7262.9
## - thinness..1.19.years             1     104.5 36820 7263.3
## - Developed                        1     157.7 36873 7267.4
## - Total.expenditure                1     160.0 36876 7267.6
## - Polio_below90                    1     187.6 36903 7269.7
## - BMI                              1     495.8 37212 7293.0
## - GDP                              1     926.3 37642 7325.4
## - Income.composition.of.resources  1    1733.0 38449 7385.0
## - Schooling                        1    4583.9 41300 7586.0
## - Adult.Mortality                  1    6319.6 43035 7701.8
## - HIV.AIDS                         1    6902.7 43619 7739.6
## 
## Step:  AIC=7255.79
## Life.expectancy ~ Adult.Mortality + infant.deaths + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + Population + thinness..1.19.years + thinness.5.9.years + 
##     Income.composition.of.resources + Schooling + Developed + 
##     Hepatitis.B_below90 + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - infant.deaths                    1       5.3 36727 7254.2
## - Population                       1       7.1 36729 7254.3
## - thinness.5.9.years               1      10.2 36732 7254.6
## <none>                                         36722 7255.8
## - Hepatitis.B_below90              1      31.1 36753 7256.2
## - Diphtheria_below90               1     100.2 36822 7261.4
## - thinness..1.19.years             1     104.5 36826 7261.8
## - Total.expenditure                1     157.6 36879 7265.8
## - Developed                        1     158.0 36880 7265.9
## - Polio_below90                    1     185.6 36907 7268.0
## - BMI                              1     490.4 37212 7291.1
## - GDP                              1     925.7 37647 7323.8
## - Income.composition.of.resources  1    1735.2 38457 7383.6
## - Schooling                        1    4578.8 41301 7584.1
## - Adult.Mortality                  1    6336.5 43058 7701.3
## - HIV.AIDS                         1    6899.4 43621 7737.8
## 
## Step:  AIC=7254.19
## Life.expectancy ~ Adult.Mortality + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + Population + thinness..1.19.years + thinness.5.9.years + 
##     Income.composition.of.resources + Schooling + Developed + 
##     Hepatitis.B_below90 + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - Population                       1       3.0 36730 7252.4
## - thinness.5.9.years               1       8.8 36736 7252.9
## <none>                                         36727 7254.2
## - Hepatitis.B_below90              1      30.6 36758 7254.5
## - Diphtheria_below90               1     100.8 36828 7259.9
## - thinness..1.19.years             1     108.1 36835 7260.5
## - Developed                        1     155.0 36882 7264.0
## - Total.expenditure                1     156.6 36884 7264.2
## - Polio_below90                    1     188.3 36915 7266.6
## - BMI                              1     489.3 37216 7289.4
## - GDP                              1     925.9 37653 7322.2
## - Income.composition.of.resources  1    1729.9 38457 7381.6
## - Schooling                        1    4601.3 41328 7584.0
## - Adult.Mortality                  1    6332.9 43060 7699.4
## - HIV.AIDS                         1    6902.1 43629 7736.3
## 
## Step:  AIC=7252.42
## Life.expectancy ~ Adult.Mortality + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + thinness..1.19.years + thinness.5.9.years + 
##     Income.composition.of.resources + Schooling + Developed + 
##     Hepatitis.B_below90 + Polio_below90 + Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## - thinness.5.9.years               1       9.3 36739 7251.1
## <none>                                         36730 7252.4
## - Hepatitis.B_below90              1      31.1 36761 7252.8
## - Diphtheria_below90               1     101.0 36831 7258.1
## - thinness..1.19.years             1     106.1 36836 7258.5
## - Developed                        1     155.6 36886 7262.3
## - Total.expenditure                1     157.0 36887 7262.4
## - Polio_below90                    1     187.2 36917 7264.7
## - BMI                              1     492.8 37223 7287.9
## - GDP                              1     925.2 37655 7320.4
## - Income.composition.of.resources  1    1738.4 38468 7380.4
## - Schooling                        1    4610.0 41340 7582.8
## - Adult.Mortality                  1    6337.0 43067 7697.8
## - HIV.AIDS                         1    6936.5 43667 7736.7
## 
## Step:  AIC=7251.14
## Life.expectancy ~ Adult.Mortality + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + thinness..1.19.years + Income.composition.of.resources + 
##     Schooling + Developed + Hepatitis.B_below90 + Polio_below90 + 
##     Diphtheria_below90
## 
##                                   Df Sum of Sq   RSS    AIC
## <none>                                         36739 7251.1
## - Hepatitis.B_below90              1      30.6 36770 7251.5
## - Diphtheria_below90               1      99.0 36838 7256.7
## - Total.expenditure                1     153.8 36893 7260.9
## - Developed                        1     154.0 36893 7260.9
## - Polio_below90                    1     187.0 36926 7263.4
## - thinness..1.19.years             1     311.8 37051 7272.9
## - BMI                              1     483.7 37223 7285.9
## - GDP                              1     921.0 37660 7318.7
## - Income.composition.of.resources  1    1744.2 38484 7379.5
## - Schooling                        1    4623.3 41363 7582.3
## - Adult.Mortality                  1    6329.7 43069 7696.0
## - HIV.AIDS                         1    6928.8 43668 7734.8
summary(clean_backward)
## 
## Call:
## lm(formula = Life.expectancy ~ Adult.Mortality + BMI + Total.expenditure + 
##     HIV.AIDS + GDP + thinness..1.19.years + Income.composition.of.resources + 
##     Schooling + Developed + Hepatitis.B_below90 + Polio_below90 + 
##     Diphtheria_below90, data = life_clean1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.1216  -2.1402  -0.1096   2.1557  16.9158 
## 
## Coefficients:
##                                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                      6.003e+01  5.102e-01 117.642  < 2e-16 ***
## Adult.Mortality                 -1.868e-02  8.507e-04 -21.956  < 2e-16 ***
## BMI                              2.708e-02  4.462e-03   6.070 1.46e-09 ***
## Total.expenditure                1.035e-01  3.024e-02   3.423 0.000629 ***
## HIV.AIDS                        -6.077e-01  2.646e-02 -22.971  < 2e-16 ***
## GDP                              5.040e-05  6.017e-06   8.375  < 2e-16 ***
## thinness..1.19.years            -9.497e-02  1.949e-02  -4.873 1.16e-06 ***
## Income.composition.of.resources  6.533e+00  5.668e-01  11.525  < 2e-16 ***
## Schooling                        7.159e-01  3.815e-02  18.764  < 2e-16 ***
## Developed                        7.763e-01  2.267e-01   3.424 0.000625 ***
## Hepatitis.B_below90              3.516e-01  2.302e-01   1.528 0.126706    
## Polio_below90                   -1.295e+00  3.432e-01  -3.774 0.000164 ***
## Diphtheria_below90              -1.002e+00  3.648e-01  -2.746 0.006076 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.624 on 2798 degrees of freedom
## Multiple R-squared:  0.8162, Adjusted R-squared:  0.8154 
## F-statistic:  1035 on 12 and 2798 DF,  p-value: < 2.2e-16
clean_forward <- step(clean_none, scope = list(lower = clean_none, upper = clean_full) ,direction = "forward")
## Start:  AIC=11987.99
## Life.expectancy ~ 1
## 
##                                   Df Sum of Sq    RSS     AIC
## + Schooling                        1    119434  80402  9430.7
## + Income.composition.of.resources  1    107856  91980  9808.9
## + Adult.Mortality                  1     98638 101199 10077.3
## + BMI                              1     62281 137556 10940.2
## + Diphtheria_below90               1     59999 139838 10986.4
## + Diphtheria_above90               1     59999 139838 10986.4
## + Polio_below90                    1     59855 139982 10989.3
## + Polio_above90                    1     59855 139982 10989.3
## + HIV.AIDS                         1     56595 143241 11054.0
## + Developed                        1     50141 149695 11177.9
## + Developing                       1     50141 149695 11177.9
## + thinness..1.19.years             1     49485 150352 11190.2
## + thinness.5.9.years               1     48828 151008 11202.4
## + GDP                              1     43164 156672 11305.9
## + Hepatitis.B_below90              1     27042 172794 11581.3
## + Hepatitis.B_above90              1     27042 172794 11581.3
## + Total.expenditure                1     14880 184957 11772.5
## + infant.deaths                    1      7696 192140 11879.6
## + Measles                          1      3712 196124 11937.3
## + Population                       1       374 199462 11984.7
## <none>                                         199836 11988.0
## 
## Step:  AIC=9430.7
## Life.expectancy ~ Schooling
## 
##                                   Df Sum of Sq   RSS    AIC
## + Adult.Mortality                  1   26227.4 54175 8322.8
## + HIV.AIDS                         1   25071.6 55331 8382.2
## + Income.composition.of.resources  1    7411.9 72990 9160.8
## + Polio_below90                    1    7162.6 73240 9170.4
## + Polio_above90                    1    7162.6 73240 9170.4
## + Diphtheria_below90               1    6606.4 73796 9191.7
## + Diphtheria_above90               1    6606.4 73796 9191.7
## + BMI                              1    5542.6 74860 9231.9
## + GDP                              1    4143.2 76259 9284.0
## + thinness.5.9.years               1    3984.5 76418 9289.8
## + thinness..1.19.years             1    3724.0 76678 9299.4
## + Hepatitis.B_below90              1    2665.8 77737 9337.9
## + Hepatitis.B_above90              1    2665.8 77737 9337.9
## + Developed                        1    2480.5 77922 9344.6
## + Developing                       1    2480.5 77922 9344.6
## + Total.expenditure                1     405.3 79997 9418.5
## + infant.deaths                    1     256.3 80146 9423.7
## <none>                                         80402 9430.7
## + Population                       1      14.3 80388 9432.2
## + Measles                          1       8.0 80394 9432.4
## 
## Step:  AIC=8322.85
## Life.expectancy ~ Schooling + Adult.Mortality
## 
##                                   Df Sum of Sq   RSS    AIC
## + HIV.AIDS                         1    8916.6 45258 7819.3
## + Polio_below90                    1    3644.8 50530 8129.1
## + Polio_above90                    1    3644.8 50530 8129.1
## + Diphtheria_below90               1    3447.4 50728 8140.0
## + Diphtheria_above90               1    3447.4 50728 8140.0
## + Income.composition.of.resources  1    3416.9 50758 8141.7
## + BMI                              1    2228.3 51947 8206.8
## + thinness..1.19.years             1    2079.7 52095 8214.8
## + GDP                              1    2070.1 52105 8215.3
## + thinness.5.9.years               1    2028.4 52147 8217.6
## + Hepatitis.B_below90              1    1380.5 52795 8252.3
## + Hepatitis.B_above90              1    1380.5 52795 8252.3
## + Developed                        1    1247.6 52927 8259.4
## + Developing                       1    1247.6 52927 8259.4
## + infant.deaths                    1     279.0 53896 8310.3
## + Total.expenditure                1     245.9 53929 8312.1
## + Measles                          1      46.2 54129 8322.4
## <none>                                         54175 8322.8
## + Population                       1      36.9 54138 8322.9
## 
## Step:  AIC=7819.34
## Life.expectancy ~ Schooling + Adult.Mortality + HIV.AIDS
## 
##                                   Df Sum of Sq   RSS    AIC
## + Income.composition.of.resources  1   3100.23 42158 7621.9
## + GDP                              1   2363.60 42895 7670.6
## + Polio_below90                    1   2228.05 43030 7679.4
## + Polio_above90                    1   2228.05 43030 7679.4
## + Diphtheria_below90               1   2177.42 43081 7682.7
## + Diphtheria_above90               1   2177.42 43081 7682.7
## + BMI                              1   1492.13 43766 7727.1
## + thinness..1.19.years             1   1375.84 43883 7734.6
## + thinness.5.9.years               1   1302.06 43956 7739.3
## + Developed                        1   1277.73 43981 7740.8
## + Developing                       1   1277.73 43981 7740.8
## + Hepatitis.B_below90              1    682.62 44576 7778.6
## + Hepatitis.B_above90              1    682.62 44576 7778.6
## + Total.expenditure                1    512.14 44746 7789.4
## + infant.deaths                    1    310.60 44948 7802.0
## + Population                       1     60.77 45198 7817.6
## + Measles                          1     46.09 45212 7818.5
## <none>                                         45258 7819.3
## 
## Step:  AIC=7621.87
## Life.expectancy ~ Schooling + Adult.Mortality + HIV.AIDS + Income.composition.of.resources
## 
##                        Df Sum of Sq   RSS    AIC
## + Polio_below90         1   1873.96 40284 7496.1
## + Polio_above90         1   1873.96 40284 7496.1
## + Diphtheria_below90    1   1786.80 40371 7502.1
## + Diphtheria_above90    1   1786.80 40371 7502.1
## + GDP                   1   1631.40 40527 7512.9
## + thinness..1.19.years  1   1154.14 41004 7545.8
## + BMI                   1   1113.22 41045 7548.7
## + thinness.5.9.years    1   1104.23 41054 7549.3
## + Developed             1    890.66 41268 7563.9
## + Developing            1    890.66 41268 7563.9
## + Total.expenditure     1    635.80 41522 7581.2
## + Hepatitis.B_below90   1    566.00 41592 7585.9
## + Hepatitis.B_above90   1    566.00 41592 7585.9
## + infant.deaths         1    378.22 41780 7598.5
## + Population            1     82.36 42076 7618.4
## + Measles               1     67.12 42091 7619.4
## <none>                              42158 7621.9
## 
## Step:  AIC=7496.06
## Life.expectancy ~ Schooling + Adult.Mortality + HIV.AIDS + Income.composition.of.resources + 
##     Polio_below90
## 
##                        Df Sum of Sq   RSS    AIC
## + GDP                   1   1530.47 38754 7389.2
## + thinness..1.19.years  1   1178.20 39106 7414.6
## + BMI                   1   1077.84 39206 7421.8
## + thinness.5.9.years    1   1056.68 39228 7423.3
## + Developed             1    828.58 39456 7439.6
## + Developing            1    828.58 39456 7439.6
## + Total.expenditure     1    579.58 39705 7457.3
## + infant.deaths         1    192.43 40092 7484.6
## + Hepatitis.B_below90   1    102.47 40182 7490.9
## + Hepatitis.B_above90   1    102.47 40182 7490.9
## + Diphtheria_below90    1     85.09 40199 7492.1
## + Diphtheria_above90    1     85.09 40199 7492.1
## + Population            1     38.20 40246 7495.4
## <none>                              40284 7496.1
## + Measles               1     21.47 40263 7496.6
## 
## Step:  AIC=7389.19
## Life.expectancy ~ Schooling + Adult.Mortality + HIV.AIDS + Income.composition.of.resources + 
##     Polio_below90 + GDP
## 
##                        Df Sum of Sq   RSS    AIC
## + thinness..1.19.years  1   1008.11 37746 7317.1
## + BMI                   1    998.77 37755 7317.8
## + thinness.5.9.years    1    875.17 37879 7327.0
## + Total.expenditure     1    541.92 38212 7351.6
## + Developed             1    338.06 38416 7366.6
## + Developing            1    338.06 38416 7366.6
## + infant.deaths         1    175.47 38578 7378.4
## + Diphtheria_below90    1     81.70 38672 7385.3
## + Diphtheria_above90    1     81.70 38672 7385.3
## + Population            1     30.19 38724 7389.0
## <none>                              38754 7389.2
## + Measles               1     19.84 38734 7389.7
## + Hepatitis.B_below90   1     17.26 38736 7389.9
## + Hepatitis.B_above90   1     17.26 38736 7389.9
## 
## Step:  AIC=7317.09
## Life.expectancy ~ Schooling + Adult.Mortality + HIV.AIDS + Income.composition.of.resources + 
##     Polio_below90 + GDP + thinness..1.19.years
## 
##                       Df Sum of Sq   RSS    AIC
## + BMI                  1    480.84 37265 7283.1
## + Total.expenditure    1    301.93 37444 7296.5
## + Developed            1    213.21 37532 7303.2
## + Developing           1    213.21 37532 7303.2
## + Diphtheria_below90   1     77.36 37668 7313.3
## + Diphtheria_above90   1     77.36 37668 7313.3
## <none>                             37746 7317.1
## + Hepatitis.B_below90  1     11.34 37734 7318.2
## + Hepatitis.B_above90  1     11.34 37734 7318.2
## + Population           1      8.98 37737 7318.4
## + thinness.5.9.years   1      1.00 37745 7319.0
## + Measles              1      0.65 37745 7319.0
## + infant.deaths        1      0.54 37745 7319.1
## 
## Step:  AIC=7283.06
## Life.expectancy ~ Schooling + Adult.Mortality + HIV.AIDS + Income.composition.of.resources + 
##     Polio_below90 + GDP + thinness..1.19.years + BMI
## 
##                       Df Sum of Sq   RSS    AIC
## + Developed            1   251.493 37013 7266.0
## + Developing           1   251.493 37013 7266.0
## + Total.expenditure    1   251.346 37013 7266.0
## + Diphtheria_below90   1    88.570 37176 7278.4
## + Diphtheria_above90   1    88.570 37176 7278.4
## <none>                             37265 7283.1
## + Hepatitis.B_below90  1    13.474 37251 7284.0
## + Hepatitis.B_above90  1    13.474 37251 7284.0
## + Population           1     5.415 37259 7284.6
## + Measles              1     4.530 37260 7284.7
## + thinness.5.9.years   1     2.353 37262 7284.9
## + infant.deaths        1     0.202 37265 7285.0
## 
## Step:  AIC=7266.02
## Life.expectancy ~ Schooling + Adult.Mortality + HIV.AIDS + Income.composition.of.resources + 
##     Polio_below90 + GDP + thinness..1.19.years + BMI + Developed
## 
##                       Df Sum of Sq   RSS    AIC
## + Total.expenditure    1   171.229 36842 7255.0
## + Diphtheria_below90   1    83.869 36929 7261.6
## + Diphtheria_above90   1    83.869 36929 7261.6
## <none>                             37013 7266.0
## + Hepatitis.B_below90  1     4.590 37009 7267.7
## + Hepatitis.B_above90  1     4.590 37009 7267.7
## + thinness.5.9.years   1     4.322 37009 7267.7
## + Population           1     4.175 37009 7267.7
## + Measles              1     2.774 37011 7267.8
## + infant.deaths        1     0.363 37013 7268.0
## 
## Step:  AIC=7254.99
## Life.expectancy ~ Schooling + Adult.Mortality + HIV.AIDS + Income.composition.of.resources + 
##     Polio_below90 + GDP + thinness..1.19.years + BMI + Developed + 
##     Total.expenditure
## 
##                       Df Sum of Sq   RSS    AIC
## + Diphtheria_below90   1    72.073 36770 7251.5
## + Diphtheria_above90   1    72.073 36770 7251.5
## <none>                             36842 7255.0
## + thinness.5.9.years   1     7.332 36835 7256.4
## + Measles              1     4.238 36838 7256.7
## + Population           1     3.848 36838 7256.7
## + Hepatitis.B_below90  1     3.722 36838 7256.7
## + Hepatitis.B_above90  1     3.722 36838 7256.7
## + infant.deaths        1     0.614 36841 7256.9
## 
## Step:  AIC=7251.48
## Life.expectancy ~ Schooling + Adult.Mortality + HIV.AIDS + Income.composition.of.resources + 
##     Polio_below90 + GDP + thinness..1.19.years + BMI + Developed + 
##     Total.expenditure + Diphtheria_below90
## 
##                       Df Sum of Sq   RSS    AIC
## + Hepatitis.B_below90  1   30.6442 36739 7251.1
## + Hepatitis.B_above90  1   30.6442 36739 7251.1
## <none>                             36770 7251.5
## + thinness.5.9.years   1    8.8795 36761 7252.8
## + Population           1    4.0503 36766 7253.2
## + Measles              1    3.7115 36766 7253.2
## + infant.deaths        1    0.3162 36770 7253.5
## 
## Step:  AIC=7251.14
## Life.expectancy ~ Schooling + Adult.Mortality + HIV.AIDS + Income.composition.of.resources + 
##     Polio_below90 + GDP + thinness..1.19.years + BMI + Developed + 
##     Total.expenditure + Diphtheria_below90 + Hepatitis.B_below90
## 
##                      Df Sum of Sq   RSS    AIC
## <none>                            36739 7251.1
## + thinness.5.9.years  1    9.3312 36730 7252.4
## + Population          1    3.5451 36736 7252.9
## + Measles             1    3.2099 36736 7252.9
## + infant.deaths       1    0.5214 36739 7253.1
summary(clean_forward)
## 
## Call:
## lm(formula = Life.expectancy ~ Schooling + Adult.Mortality + 
##     HIV.AIDS + Income.composition.of.resources + Polio_below90 + 
##     GDP + thinness..1.19.years + BMI + Developed + Total.expenditure + 
##     Diphtheria_below90 + Hepatitis.B_below90, data = life_clean1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.1216  -2.1402  -0.1096   2.1557  16.9158 
## 
## Coefficients:
##                                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                      6.003e+01  5.102e-01 117.642  < 2e-16 ***
## Schooling                        7.159e-01  3.815e-02  18.764  < 2e-16 ***
## Adult.Mortality                 -1.868e-02  8.507e-04 -21.956  < 2e-16 ***
## HIV.AIDS                        -6.077e-01  2.646e-02 -22.971  < 2e-16 ***
## Income.composition.of.resources  6.533e+00  5.668e-01  11.525  < 2e-16 ***
## Polio_below90                   -1.295e+00  3.432e-01  -3.774 0.000164 ***
## GDP                              5.040e-05  6.017e-06   8.375  < 2e-16 ***
## thinness..1.19.years            -9.497e-02  1.949e-02  -4.873 1.16e-06 ***
## BMI                              2.708e-02  4.462e-03   6.070 1.46e-09 ***
## Developed                        7.763e-01  2.267e-01   3.424 0.000625 ***
## Total.expenditure                1.035e-01  3.024e-02   3.423 0.000629 ***
## Diphtheria_below90              -1.002e+00  3.648e-01  -2.746 0.006076 ** 
## Hepatitis.B_below90              3.516e-01  2.302e-01   1.528 0.126706    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.624 on 2798 degrees of freedom
## Multiple R-squared:  0.8162, Adjusted R-squared:  0.8154 
## F-statistic:  1035 on 12 and 2798 DF,  p-value: < 2.2e-16

Unfortunately, the adj. R-squared significantly decreased from 0.823 to 0.8154. This is not what we anticipated. We will therefore maintain the original data source along with any outliers.

Transform the Data

Let’s attempt to use Log to alter the data. We will only use the model’s internal variables as we have previously determined that “model backward” fits the data the best.

log_life <- lm(formula = log1p(Life.expectancy) ~ log1p(Adult.Mortality) + log1p(Measles) + log1p(BMI)    +log1p(Total.expenditure) + log1p(HIV.AIDS) + log1p(GDP) + log1p(thinness..1.19.years) + 
    log1p(Income.composition.of.resources) + log1p(Schooling) + log1p(Developed) + 
    log1p(Polio_below90) + log1p(Diphtheria_below90), data = life_exp1)
summary(log_life)
## 
## Call:
## lm(formula = log1p(Life.expectancy) ~ log1p(Adult.Mortality) + 
##     log1p(Measles) + log1p(BMI) + log1p(Total.expenditure) + 
##     log1p(HIV.AIDS) + log1p(GDP) + log1p(thinness..1.19.years) + 
##     log1p(Income.composition.of.resources) + log1p(Schooling) + 
##     log1p(Developed) + log1p(Polio_below90) + log1p(Diphtheria_below90), 
##     data = life_exp1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.50957 -0.03026  0.00100  0.03204  0.23753 
## 
## Coefficients:
##                                          Estimate Std. Error t value Pr(>|t|)
## (Intercept)                             4.1002713  0.0148532 276.053  < 2e-16
## log1p(Adult.Mortality)                 -0.0100412  0.0011572  -8.677  < 2e-16
## log1p(Measles)                         -0.0030335  0.0003680  -8.243 2.50e-16
## log1p(BMI)                              0.0048344  0.0016750   2.886  0.00393
## log1p(Total.expenditure)                0.0044435  0.0030113   1.476  0.14016
## log1p(HIV.AIDS)                        -0.0953165  0.0017230 -55.321  < 2e-16
## log1p(GDP)                              0.0105913  0.0007734  13.694  < 2e-16
## log1p(thinness..1.19.years)            -0.0133416  0.0020399  -6.540 7.21e-11
## log1p(Income.composition.of.resources)  0.1684808  0.0118894  14.171  < 2e-16
## log1p(Schooling)                        0.0374174  0.0042277   8.850  < 2e-16
## log1p(Developed)                        0.0310823  0.0051193   6.072 1.43e-09
## log1p(Polio_below90)                   -0.0199477  0.0078016  -2.557  0.01061
## log1p(Diphtheria_below90)              -0.0236010  0.0077920  -3.029  0.00248
##                                           
## (Intercept)                            ***
## log1p(Adult.Mortality)                 ***
## log1p(Measles)                         ***
## log1p(BMI)                             ** 
## log1p(Total.expenditure)                  
## log1p(HIV.AIDS)                        ***
## log1p(GDP)                             ***
## log1p(thinness..1.19.years)            ***
## log1p(Income.composition.of.resources) ***
## log1p(Schooling)                       ***
## log1p(Developed)                       ***
## log1p(Polio_below90)                   *  
## log1p(Diphtheria_below90)              ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.05847 on 2925 degrees of freedom
## Multiple R-squared:  0.8372, Adjusted R-squared:  0.8366 
## F-statistic:  1254 on 12 and 2925 DF,  p-value: < 2.2e-16

The Adj. R-Squared is even larger than the “model backward,” which is good.

bc <- MASS::boxcox(model_backward) # boxcox the model_backward

lambda <- bc$x[which.max(bc$y)] # choose the best lambda

powerTransform <- function(y, lambda1, lambda2 = NULL, method = "boxcox") {

  boxcoxTrans <- function(x, lam1, lam2 = NULL) {

    # if we set lambda2 to zero, it becomes the one parameter transformation
    lam2 <- ifelse(is.null(lam2), 0, lam2)

    if (lam1 == 0L) {
      log(y + lam2)
    } else {
      (((y + lam2)^lam1) - 1) / lam1
    }
  }

  switch(method
         , boxcox = boxcoxTrans(y, lambda1, lambda2)
         , tukey = y^lambda1
  )
}

boxcox_life <- lm(powerTransform(Life.expectancy, lambda) ~ Adult.Mortality + Measles + BMI + 
    Total.expenditure + HIV.AIDS + GDP + thinness..1.19.years + 
    Income.composition.of.resources + Schooling + Developed + 
    Polio_below90 + Diphtheria_below90, data = life_exp1)

summary(boxcox_life)
## 
## Call:
## lm(formula = powerTransform(Life.expectancy, lambda) ~ Adult.Mortality + 
##     Measles + BMI + Total.expenditure + HIV.AIDS + GDP + thinness..1.19.years + 
##     Income.composition.of.resources + Schooling + Developed + 
##     Polio_below90 + Diphtheria_below90, data = life_exp1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1149.6  -161.1    -6.4   162.2  1245.7 
## 
## Coefficients:
##                                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                      1.696e+03  3.538e+01  47.930  < 2e-16 ***
## Adult.Mortality                 -1.175e+00  5.288e-02 -22.228  < 2e-16 ***
## Measles                         -8.590e-04  4.467e-04  -1.923  0.05457 .  
## BMI                              2.166e+00  3.266e-01   6.634 3.88e-11 ***
## Total.expenditure                5.698e+00  2.169e+00   2.627  0.00866 ** 
## HIV.AIDS                        -2.738e+01  1.160e+00 -23.611  < 2e-16 ***
## GDP                              4.209e-03  4.395e-04   9.577  < 2e-16 ***
## thinness..1.19.years            -6.261e+00  1.416e+00  -4.422 1.02e-05 ***
## Income.composition.of.resources  4.944e+02  4.123e+01  11.994  < 2e-16 ***
## Schooling                        5.159e+01  2.753e+00  18.740  < 2e-16 ***
## Developed                        8.503e+01  1.660e+01   5.122 3.23e-07 ***
## Polio_below90                   -9.153e+01  2.466e+01  -3.712  0.00021 ***
## Diphtheria_below90              -6.222e+01  2.474e+01  -2.515  0.01196 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 267.5 on 2925 degrees of freedom
## Multiple R-squared:  0.8216, Adjusted R-squared:  0.8209 
## F-statistic:  1123 on 12 and 2925 DF,  p-value: < 2.2e-16

The Adj. R-Squared is smaller than the “model_backward”, we will not consider the box-cox transformation, and will use log transformation as our new model.

Second Normality Test

Plot Residuals on Histogram
hist(log_life$residuals, breaks = 20)

The Residuals seems has a better distribution at center.

Plot Residuals on QQPlot
plot(log_life, which = 2)

The residuals under -2 and over 2 fell way above/below the center line. Seems still not following the normal distribution.

Shapiro Test
library(olsrr)
ols_test_normality(log_life)
## -----------------------------------------------
##        Test             Statistic       pvalue  
## -----------------------------------------------
## Shapiro-Wilk              0.9554         0.0000 
## Kolmogorov-Smirnov        0.055          0.0000 
## Cramer-von Mises         876.6773        0.0000 
## Anderson-Darling         17.1011         0.0000 
## -----------------------------------------------

Based on Shapiro-Wilk, Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling normality test, the p-value < 0.05 implying that the distribution of the data are significantly different from normal distribution. Therefore, we need to do some adjustment to data.

Homoscedasticity

Plot Fitted vs Residuals
par(mfrow=c(2,1))
plot(log_life, which = c(1:4))

The error seems not following particular pattern, by visual plot analysis.

Breusch-Pagan Test
library(lmtest)
bptest(log_life)
## 
##  studentized Breusch-Pagan test
## 
## data:  log_life
## BP = 188.84, df = 12, p-value < 2.2e-16
ols_test_breusch_pagan(log_life)
## 
##  Breusch Pagan Test for Heteroskedasticity
##  -----------------------------------------
##  Ho: the variance is constant            
##  Ha: the variance is not constant        
## 
##                        Data                        
##  --------------------------------------------------
##  Response : log1p(Life.expectancy) 
##  Variables: fitted values of log1p(Life.expectancy) 
## 
##          Test Summary           
##  -------------------------------
##  DF            =    1 
##  Chi2          =    344.8674 
##  Prob > Chi2   =    5.557104e-77

Using 2 different function to test the homocedasticity, we still get conclusion that the residuals variance is not constant.

Multicollinearity Test
vif(log_life)
##                 log1p(Adult.Mortality)                         log1p(Measles) 
##                               1.234068                               1.235571 
##                             log1p(BMI)               log1p(Total.expenditure) 
##                               1.364789                               1.148442 
##                        log1p(HIV.AIDS)                             log1p(GDP) 
##                               1.495744                               1.693346 
##            log1p(thinness..1.19.years) log1p(Income.composition.of.resources) 
##                               1.817267                               2.600192 
##                       log1p(Schooling)                       log1p(Developed) 
##                               2.234320                               1.556913 
##                   log1p(Polio_below90)              log1p(Diphtheria_below90) 
##                               6.139776                               6.142181

Given that the results of the vif test are less than 10, we may conclude that there is no correlation between any of the independent variables.

Since there is no p-value greater than 0.05, all chosen variables exhibit linear correlation with the dependent variable.

Conclusion

Based on the Adjusted R-Squared value, Error Value, and Pass 2 of 4 Assumption Check, which is the Multicollinearity and Linearity Test, the linear model appears to be able to predict Life Expectancy. However, homocedasticity and normality do not produce the desired outcome. Even though the residuals plot appears to follow the homocedasticity and normal distribution principles when viewed visually, the results of the statistical test are different.

The linear association between Life Expectancy and the chosen independent variables can be explained using the linear model. It is strongly advised to look at the outliers pattern if we still want to apply this model on the new set of Life.expectancy data because it is highly sensitive to outliers (which pretty massively occurred in this data and taking it out is not a smart option).